Download spin exchange optical pumping of neon and its applications

S PIN EXCHANGE O PTICAL PUMPING OF N EON
AND ITS
A PPLICATIONS
R AJAT K. G HOSH
A D ISSERTATION
P RESENTED TO THE FACULTY
OF
IN
P RINCETON U NIVERSITY
C ANDIDACY FOR THE D EGREE
OF
D OCTOR OF P HILOSOPHY
R ECOMMENDED FOR A CCEPTANCE
BY THE
D EPARTMENT OF
P HYSICS
A DVISER : M ICHAEL V. R OMALIS
S EPTEMBER 2009
c Copyright by Rajat K. Ghosh, 2009.
All Rights Reserved
Abstract
Hyperpolarized noble gases are used in a variety of applications including medical diagnostic lung imaging, tests of fundamental symmetries, spin filters, atomic
gyroscopes, and atomic magnetometers. Typically 3 He is utilized because large
3 He
polarizations on the order of 80% can be achieved. This is accomplished by
optically pumping an alkali vapour which polarizes a noble gas nucleus via spin
exchange optical pumping.
One hyperpolarized noble gas application of particular importance is the K3 He
co-magnetometer. Here, the alkali atoms optically pump a diamagnetic noble
gas. The magnetic holding field for the alkali and noble gas is reduced until both
species are brought into hybrid magnetic resonance. The co-magnetometer exhibits many useful attributes which make it ideal for tests of fundamental physics,
such as insensitivity to magnetic fields.
The co-magnetometer would demonstrate increased sensitivity by replacing
3 He
with polarized
21 Ne
gas. Tests of CPT violation using co-magnetometers
would be greatly improved if one utilizes polarized
21 Ne
gas. The sensitivity of
the nuclear spin gyroscope is inversely proportional to the gyromagnetic ratio of
the noble gas. Switching to neon would instigate an order of magnitude gain in
sensitivity over 3 He.
In order to realize these applications the interaction parameters of
21 Ne
with
alkali metals must be measured. The spin-exchange cross section σse , and magnetic field enhancement factor κ0 are unknown, and have only been theoretically
calculated. There are no quantitative predictions of the neon-neon quadrupolar
relaxation rate Γquad .
In this thesis I test the application of a K-3 He co-magnetometer as a navigational
gyroscope. I discuss the advantages of switching the buffer gas to 21 Ne. I discuss
the feasibility of utilizing polarized 21 Ne for operation in a co-magnetometer, and
iii
construct a prototype
21 Ne
co-magnetometer. I investigate polarizing
21 Ne
with
optical pumping via spin exchange collisions and measure the spin exchange rate
coefficient of K and Rb with Ne to be 2.9 × 10−20 cm3 /s and 0.81 × 10−19 cm3 /s.
We measure the magnetic field enhancement factor κ0 to be 30.8 ± 2.7, and 35.7 ±
3.7 for the K-Ne, and the Rb-Ne pair. We measure the quadrupolar relaxation
coefficient to be 214 ± 10 Amagat·s. Furthermore the spin destruction cross section
of Rb, and K with 21 Ne is measured to be 1.9 × 10−23 cm2 and 1.1 × 10−23 cm2 .
iv
Acknowledgements
I would first like to thank my advisor Michael Romalis for his help in the lab. His
passion to study fundamental physics is what drives the entire lab group. Without
his assistance and guidance this work would not have been possible. Of special
note is his availability to discuss the underlying physics so that one can gain a
deeper view of the atomic processes which we study.
The projects described herein have greatly benefited from the fruitful collaboration with my colleagues in the Romalis lab. These projects could not have been
accomplished by me alone. Although I could never fully express my appreciation
and admiration for my fellow colleagues I would still like to take the time to thank
them each individually. First I would like to thank Tom Kornack. He built the
first generation CPT violation experiment. Without his help I would not have been
able to get any gyroscope data. I would also like to thank Γιὼγos Bασιλὰκηs, and
Sylvia Smullin for their help with the scalar magnetometer experiment. I would
also like to thank Vishal Shah for his help with regards to the spin exchange rate
measurements. I really appreciate all the time you have taken to discuss all things
atomic physics.
This work has benefited from the support of the many staff members in the
physics department. I would like to thank Bill Dix for his help in machining the
parts of my experiment when I was unable to do so. I would also like to thank
Ted Lewis in the metal stockroom for all of his help. I would be remiss if I did not
thank Mike Peloso for all his help teaching me how to machine in the student shop
and always suggesting alternative designs, or easier ways in which to machine my
designs. I will miss our talks, or more accurately shouts, about guitars, and music
theory over the commercial lathe and drill press.
I would like to thank Mary Santay, Barbara Grunwerg, Kathy Warren, John
Washington, from the purchasing and receiving department. You really make it a
v
pleasure to come visit the A level. I would also like to thank Claude Champagne
of the purchasing department for always taking the time to try and make all of us
smile. I would also like to thank Regina Savadge, Ellen Webster Synakowski, and
Mary Delorenzo as the vanguards of the atomic physics group. I would also like to
thank Mike Souza for the glass cells which only he could so expertly craft. I would
like to give a special thanks to Laurel Lerner for all her help in all things, especially
near the end of my time here.
During my tenure here I have spent much of my time with my fellow grad students in the deepest darkest Jadwin. You have been both great colleagues, and
even better friends. First I must thank Scott Seltzer for our many tea time conversations about not only atomic physics, but politics, cinema, and the finer points
of life in general. I do miss our discussions about british comedy, and the beauty
of a proper Jewish Deli and the perfect matza ball soup. I would like to thank
the enthusiasm Justin Brown has shown over the years. If only we could all have
the energy you do I think the World wouldn’t need coffee. As for my good friend
Γιὼγos Bασιλὰκηs I will miss your contagious love of science, and being able to
debate the biggest proponent of the Greek culture. My only hope is that they do
not one day find the work of my thesis had already been scribed on one of Aristotle’s tablets. I would also like to thank Ranjit Chima. Friends like you are very
rare. I think life in Princeton would have not been nearly as fun had you not been
here.
I would also like to acknowledge all the past members of the atomic physics
group from whom I have learnt a great deal over the years including Micah Ledbetter, Hui Xia, Kiwoong Kim, Charles Sule, Andre Baranga, Oleg Polyakov, Parker
Meares, Scott Seltzer and Dan Hoffman. I will miss my conversations with Dan
about all things, and with Kiwoong about when it is appropriate to acknowledge
commoners.
vi
I would like to give thanks to my readers, and my committee members for taking the time to read my thesis and giving useful suggestions for the improvement
of this manuscript. I am also grateful to Scott Seltzer for his careful reading of the
first draft of my thesis.
Finally and most importantly I would like to thank all of my family. Without
you none of this would have been possible. I would like to thank my sister Sheila
for all her support. Your dedication to your own work encourages me to always
stay positive and persevere. You have always listened to me when I needed comforting, and cheered me when I need cheering. I don’t think I could have gone
through this without you. I have to give a special thanks to my parents. Without your guidance none of what I have accomplished would have been possible.
I would like to thank my mother for her constant encouragement, her support,
and her unwavering belief in me. Without you pushing me when I needed to be
pushed, and encouraging me when I needed to be encouraged I would not have
made it this far. I must also thank my father for his constant help in both discussing all thing physics, and in his advice. From the time I was little you have
always helped me in my education, and taught me a great deal. Your constant
encouragement has meant all the difference to me. To all of my family I owe you a
debt which I can never acknowledge enough, or ever justly repay. You inspire me.
vii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
Table of Contents
x
List of Tables
xi
List of Figures
xiv
1
Introduction
1
2
Background
7
2.1
Optical Pumping background . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.1
Atomic Energy Levels . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.2
Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3
Effects of Pumping Rate and resonance Lineshape on Optical
Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2
2.1.4
Evolution of Alkali polarization due to Optical Pumping . . . 17
2.1.5
Dynamics of polarized alkali at low magnetic fields . . . . . . 20
Spin Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1
Spin Exchange Collisions . . . . . . . . . . . . . . . . . . . . . 24
2.2.2
Spin Destruction Collisions . . . . . . . . . . . . . . . . . . . . 29
2.2.3
Diffusion wall collisions, and Magnetic field Gradients . . . . 30
viii
2.3
2.4
2.5
3
4
Monitoring polarized Alkali . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.1
Optical Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.2
Light Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Coupled Spin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4.1
Optical Pumping of Noble Gas . . . . . . . . . . . . . . . . . . 42
2.4.2
Interaction of polarized alkali with polarized noble gas . . . . 44
Manipulation of polarized noble gas spins, and Magnetic shielding . 48
2.5.1
Adiabatic Fast Passage . . . . . . . . . . . . . . . . . . . . . . . 49
2.5.2
Nuclear Magnetic Resonance . . . . . . . . . . . . . . . . . . . 52
2.5.3
Magnetic Shielding . . . . . . . . . . . . . . . . . . . . . . . . . 55
Nuclear Spin Gyroscope
59
3.1
Co-magnetometer Gyroscope Implementation and behaviour . . . . 60
3.2
Effect of Experimental Imperfections on Gyroscope Performance . . 66
3.3
Zeroing the Co-magnetometer Gyroscope . . . . . . . . . . . . . . . . 67
3.4
Co-magnetometer Gyroscope Sensitivity . . . . . . . . . . . . . . . . . 70
Initial tests of an alkali-Neon co-magnetometer
75
4.1
Magnetometer setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2
Neon Polarization Measurements and Preliminary Neon Co-Magnetometer
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3
Influence of Quadrupole collisions in Polarizing neon nuclei . . . . . 77
4.3.1
4.4
T1 measurement of Neon . . . . . . . . . . . . . . . . . . . . . 80
Improving Magnetometer Sensitivity . . . . . . . . . . . . . . . . . . . 84
4.4.1
Removing Birefringence and false Faraday Rotation signals . 84
4.4.2
Controlling and Monitoring the Laser stability . . . . . . . . . 85
4.4.3
Miniaturization of Gyroscope . . . . . . . . . . . . . . . . . . . 86
4.4.4
Alternate methods to heat Cell, and remove Convection noise 87
ix
5
6
Measurement of parameters for Polarizing Ne with K or Rb metal
91
5.1
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.1
NMR detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.2
Electron Paramagnetic Resonance Shift . . . . . . . . . . . . . 99
5.2.3
Alkali Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.4
Alkali Polarization Decay Constant measurement . . . . . . . 103
5.2.5
Back Polarization measurement . . . . . . . . . . . . . . . . . . 104
5.2.6
Alkali density measurement . . . . . . . . . . . . . . . . . . . . 105
5.3
Fermi Contact interaction κ0 Results . . . . . . . . . . . . . . . . . . . 107
5.4
Results of neon quadrupolar relaxation Γquad measurement . . . . . . 107
5.5
Spin exchange Rate coefficient Results . . . . . . . . . . . . . . . . . . 108
5.6
Measurement of Spin destruction cross-sections of neon with Rb and K110
5.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Feasibility of utilizing 21 Ne in a co-magnetometer
6.1
114
Effects of Light Propogation and alkali relaxation on Rb-Ne co-magnetometer
simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.2
Simulation of Noble gas relaxation . . . . . . . . . . . . . . . . . . . . 117
6.3
Noise mechanisms in a Rb-Ne co-magnetometer . . . . . . . . . . . . 118
6.4
7
6.3.1
Spin Projection Noise . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3.2
Photon Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . 120
Results of Rb-Ne co-magnetometer simulation . . . . . . . . . . . . . 122
Conclusions and future work
125
A Properties of Ne21
129
x
List of Tables
2.1
Line broadening and shift of K in various Gases . . . . . . . . . . . . 16
2.2
Spin Destruction Cross Sections . . . . . . . . . . . . . . . . . . . . . . 30
3.1
Gyroscope performance comparison . . . . . . . . . . . . . . . . . . . 74
5.1
K-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . . 110
5.2
Rb-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . 110
5.3
Fermi Contact Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4
Rb-Ne and K-Ne spin destruction cross sections. . . . . . . . . . . . . 112
7.1
K-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . . 125
7.2
Rb-Ne spin exchange parameters . . . . . . . . . . . . . . . . . . . . . 126
7.3
Fermi Contact Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.4
Rb-Ne and K-Ne spin destruction cross sections. . . . . . . . . . . . . 126
A.1 Properties of Ne21 relevant for optical pumping . . . . . . . . . . . . 130
xi
List of Figures
1.1
Basic operation of an Atomic magnetometer . . . . . . . . . . . . . . .
3
2.1
Alkali metal energy level diagram . . . . . . . . . . . . . . . . . . . .
9
2.2
Optical pumping of the electron spin of an alkali atom . . . . . . . . . 12
2.3
Ground-state Zeeman level splitting . . . . . . . . . . . . . . . . . . . 26
2.4
Spin-exchange collisions can cause atoms to switch hyperfine levels . 26
2.5
Spin-temperature distribution . . . . . . . . . . . . . . . . . . . . . . . 28
2.6
Principle of optical rotation . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7
Branching ratios for the D1 and D2 transitions . . . . . . . . . . . . . 35
2.8
Methods for detecting optical rotation . . . . . . . . . . . . . . . . . . 36
2.9
Methods for detecting optical rotation . . . . . . . . . . . . . . . . . . 37
2.10 Polarized Noble gas screens transverse fields . . . . . . . . . . . . . . 48
2.11 Effective magnetic field in the rotating frame . . . . . . . . . . . . . . 51
2.12 NMR tip of atoms with spin ~S . . . . . . . . . . . . . . . . . . . . . . . 54
2.13 Affinity of Magnetic fields line for Magnetic Shields . . . . . . . . . . 57
3.1
Schematic of the co-magnetometer implemented as a gyroscope . . . 61
3.2
Side view of the gyroscope configuration for the co-magnetometer . 62
3.3
In-situ calibration of non-contact displacement sensors. . . . . . . . . 62
3.4
Comparison of co-magnetometer gyroscope signal to displacement
sensor signal with no free parameters . . . . . . . . . . . . . . . . . . 63
xii
3.5
noise spectrum of the comagnetometer gyroscope . . . . . . . . . . . 63
3.6
Suppression of an applied magnetic field gradient by the co-magnetometer
compared to that of a non-compensating magnetometer . . . . . . . . 64
3.7
The co-magnetometer suppressing magnetic fields . . . . . . . . . . . 65
3.8
Response of co-magnetometer to a magnetic field transient . . . . . . 65
3.9
Long term drift of gyroscope . . . . . . . . . . . . . . . . . . . . . . . 72
4.1
Experimental setup of Neon Magetometer . . . . . . . . . . . . . . . . 76
4.2
T2 time of ≈ 14minutes for Neon polarization when operating away
from the compensation point in the co-magnetometer configuration.
4.3
77
Compensation behaviour of the K-Ne comagnetometer to an externally applied magnetic transient field . . . . . . . . . . . . . . . . . . 78
4.4
T1 of 105 minutes for a 1.6atm cell of Ne at 170C◦ . . . . . . . . . . . . 79
4.5
Theoritical simulation of the noble gas spin for positive gain κ . . . . 83
4.6
Theoritical simulation of the noble gas spin for negative gain κ . . . . 83
4.7
Magnetic field homogeneity for a 3cm×3cm region in magetic shields 87
4.8
Silvered oven holding Boron-Nitride housing for Pyrex cell . . . . . . 89
5.1
Experimental Setup for measuring spin exchange parameters of K,
and Rb with Ne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2
Representative NMR signal of polarized 21 Ne gas . . . . . . . . . . . 99
5.3
Representative EPR shifts after 2 hours of polarization . . . . . . . . . 101
5.4
Determination of Alkali polarization via RF sweep over Zeeman levels103
5.5
Potassium Polarization decay as a result of Pump beam being manually chopped . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.6
Absolute Potassium Back polarization as Neon is flipped via Adiabatic fast passage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.7
Neon Relaxation as a function of cell pressure. . . . . . . . . . . . . . 108
xiii
5.8
Absolute neon polarization as function of time for determination of
Spin Exchange Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.9
Scatter in Spin exchange rate measurements for K-Ne . . . . . . . . . 111
5.10 Scatter in Spin exchange rate measurements for Rb-Ne . . . . . . . . 111
6.1
Absolute Rb polarization as function of propagation distance through
cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
xiv
Chapter 1
Introduction
Since the time of the first compass Man has been intrigued by the properties of
magnetic fields. This fascination only increased with the birth of modern electronics. However the reach of magnetism extends well beyond that of electronics. The
duality of the theory of magnetism and electricity proved to be a fundamental realization when special relativity was utilized to unify them. Even light has been
shown to be a form of electromagnetic radiation. The properties of magnetic fields
have been used for navigational purposes, and as a means to unify seemingly dissimilar forces in physics for over a century.
In the modern era one finds even more applications of magnetic fields. A few
of these include medical imaging, detection of explosives, generation of electricity,
tests of fundamental symmetries, and maglev trains. Therefore, it should be no
surprise that the ability to precisely and accurately measure magnetic fields with
as much sensitivity as possible is of prime importance.
During the past century scientists have continually sought to improve the sensitivity of magnetic field measurements. This began with Hall probes, fluxgates,
and ultimately in superconducting quantum interference devices (SQUIDS). For
the past few decades these SQUIDS have been the state of the art with respect to
1
measuring magnetic fields. They exhibit sensitivity on the order of 1fT when operating in an environment with superconducting shields (Clarke & Braginski (2004)).
However this decade has seen the re-emergence of spin precession based atomic
magnetometers with improved sensitivity which now surpass that of SQUIDS.
In 1957 Dehmelt (Dehmelt (1957)) proposed observing the precession of alkali
spins in order to measure the strength of a magnetic field. This was carried out experimentally the same year by Bell and Bloom (Bell & Bloom (1957)). These works
were aided by the contribution of Kastler who created a technique to produce significant population changes from free atoms in the ground state (Kastler (1957)).
For this work he was awarded the 1966 Nobel prize in physics.
Most modern day spin precession atomic magnetometers operate on the same
underlying principles as those used by Bell, Bloom, and Kastler. Spins are polarized by optically pumping with a laser beam, and allowed to precess in a magnetic
field according to Larmor precession. See fig.1.1.
~ = γ~B
ω
(1.1)
Here the frequency of precession is directly proportional to the strength of the
magnetic field. This precession can be detected via optical rotation of the plane of
polarization of a linearly polarized probe laser beam. The plane of polarization of
the probe laser becomes rotated as it interacts with a magnetically polarized sample. This leads to a simple way in which to measure the alkali precession frequency,
and hence the magnetic field.
Normally the atoms one uses to measure precession are alkali atoms. This is for
multiple reasons. The most important of which is that because alkali only have one
valence electron, they are simple spin systems. That is their total spin can be characterized by the sum of the nuclear spin, and valence electron spin. Furthermore
2
B
Pump Beam
ω
F
Probe Beam
Figure 1.1: For operation of a magnetometer alkali atoms are polarized via circular
polarized pump beam. A probe beam measures the orientation of the alkali polarization via Faraday rotation as the atoms precess due to a magnetic field. The
frequency of precession can be used to determine the strength of the magnetic field.
3
the ground state is spherically symmetric. This leads to long lived ground state polarization lifetimes. Additionally the alkali atoms can be easily interrogated with
lasers via a strong optical transition. This enables us to pump the majority of the
alkali into specific states so that their magnetic moments are all parallel.
In order to pump the alkali metal in its gas state, we heat the metal sample
and rely on the saturated vapour pressure to supply the vapour to be optically
pumped. The sensitivity of the spin precession magnetometer is characterized by
the time for which the spins can precess coherently.
1
γT2
∆B =
(1.2)
Where T2 is the transverse coherence time of the optically pumped vapour. Quantum mechanically we can interpret this spin precession as the measurement of the
splitting of the Zeeman levels of the alkali, in this case in its ground state. The
magnetic linewidth can be expressed as
∆B =
∆ω
γ
(1.3)
Thus to optimize the sensitivity of the magnetometer, we maximize the spin precession coherence time, and effectively minimize the linewidth of the Zeeman resonance line. Therefore, we can express the magnetometer sensitivity δB in terms
of the Zeeman level linewidth ∆B, and the signal to noise ratio of the Zeeman
resonance.
δB =
∆B
SNR
(1.4)
When the alkali atoms collide with the cell walls they depolarize. In order to increase the polarization lifetime one normally introduces a buffer gas into the cell.
One can coat the cell walls with either parrafin, or silane coatings as these reduce
4
the alkali depolarization rate upon collision. The buffer gas densities are typically
105 − 106 times greater than the alkali density under normal operating conditions.
Typically inert noble gases are utilized for this purpose. In fact if the noble gas
utilized has a nonzero nuclear spin one can transfer some of the alkali electron
spin polarization to the noble gas nuclear spin via an interaction known as spin
exchange collisions. This can be used to construct co-magnetometers, which are a
type of magnetometer containing a polarized noble gas species.
The strong D1 transitions of the alkali can easily be accessed by modern solid
state diode lasers. These lasers can easily be tuned to pump the alkali vapour,
and are quite stable in frequency. The preferred diode laser is a DFB (distributed
feedback) laser. These have reflection gratings etched directly on the diode surface which result in operation with very stable frequency. This frequency stability
makes them ideal for measuring the alkali polarization as a linearly polarized detuned probe beam.
The sensitivity to polarization of the probe beam due to spin projection noise
varies inversely as the number of alkali in the cell. This effect will be discussed
later in this text.
1
∆Sx ∝ √
N
(1.5)
One can increase this by either increasing the vapour pressure of the alkali, by increasing the temperature, or by increasing the path length of the vapour cell. If
one increases the density of the cell the alkali interact with themselves via spin exchange collisions which broaden the Zeeman resonance. For years this had been
the limiting mechanism in improving atomic magnetometer sensitivity. Typically
cells had been constructed with large volume, with low density, at room temperature in order to minimize the reduction in polarization lifetime due to spin exchange collisions.
In 2002 Allred et al. (2002) experimentally demonstrated a magnetometer which
5
operated with small measurement volume, near zero field which did not experience broadening of the Zeeman resonance due to spin exchange collisions. It does
so by operating in a spin exchange relaxation free regime (SERF) which was first
proposed by Happer & Tam (1977). Many of the magnetometers described in this
thesis are based on this effect.
In this thesis we describe the potential application of enriched neon as a buffer
gas. We also demonstrate the application of a potassium-helium co-magnetometer
as a sensitive gyroscope. We show that the sensitivity of this gyroscope can be
further improved if we switch the buffer gas from helium to neon.
This potassium-helium co-magnetometer was originally used for tests of Lorentz
violation, and CPT invariance. CPT (charge parity time reversal) is a discrete symmetry of the universe. Breaking of this symmetry would lead to a violation of
Lorentz symmetry. While these symmetries are conserved in the Standard Model
they may be broken in a more fundamental underlying theory. There exist theories of quantum gravity which have been shown to violate Lorentz symmetry in
some way. Switching to a neon buffer gas will also increase of the sensitivity of the
co-magnetometer to these effects.
However many of the parameters governing the interaction between potassium, or rubidium with enriched 21 Ne have not been measured. The spin exchange
rate between the transfer of polarization of the alkali to the noble gas nuclei has
not been measured. Neither has the Fermi contact interaction between these pairs
of atoms. In order to design and construct an optimized magnetometer one must
have a knowledge of these parameters. We measure them here.
We construct the first
21 Ne
co-magnetometer and study its behaviour. And
finally we discuss what future work must be carried out in order to use
tests of fundamental symmetries.
6
21 Ne
for
Chapter 2
Background
In order to understand the experiments involving alkali metal-neon optical pumping it would be prudent to first review the physical processes pertaining to optical
pumping. We begin by discussing the basics of polarizing alkali via optical pumping. We discuss the properties of the energy levels relevant to optical pumping as
well as the effects of line broadening on optical pumping. We also briefly discuss
utilizing polarized alkali to measure magnetic fields.
The process of optical pumping does not polarize the alkali to unity. In the next
section we discuss the relaxation mechanisms which limit the alkali polarization.
Relaxation mechanisms of particular interest include spin relaxation due to spin
exchange collisions, spin destruction collisions, and magnetic field gradients.
Additionally we discuss the means by which the alkali polarization is interrogated. This includes a number of schemes based on optical rotation utilizing
polarimeters. We also investigate spurious signals due to the effects of light shifts.
We discuss the consequences of adding diamagnetic noble gas to the vicinity of
polarized alkali atoms. We investigate the transfer of polarization from the alkali
spin to the noble gas nuclei via spin exchange optical pumping. We describe the
response of the coupled system and the dynamics of an atomic co-magnetometer.
7
Finally we discuss useful techniques with which to manipulate the polarized noble
gas spins, such as adiabatic fast passage, and nuclear magnetic resonance.
2.1
Optical Pumping background
Optical pumping is a technique which can be used to polarize the spins of an
atomic species through application of laser light. The most popular atomic species
to utilize during optical pumping experiments are the alkali atoms. Typically the
transitions utilized to optically pump the alkali are the D1 or D2 doublets. In this
section we discuss the basic principles of optical pumping. We discuss the properties which make alkali atoms the preferred species for many optical pumping
experiments. We investigate the properties of the alkali atoms energy levels which
are relevant for optical pumping and ways to manipulate these levels to produce a
long lived ground state alkali polarization. Finally we discuss the dynamics of optical pumping alkali in terms of the optical pumping rate, and the spin relaxation
rate.
2.1.1 Atomic Energy Levels
Alkali metals are a convenient choice for optical pumping because they possess
a single unpaired valence electron. The spectroscopic properties are well approximated by ignoring the interaction of the filled electron sublevels and concentrating
on the valence electron, and its interaction with the atomic nucleus. As such the
wavefunctions describing the energy levels are well described by total angular momentum quantum numbers of the valence electron spin and the nuclear spin.
Consider the ground state S shell of the alkali atom. The spin of the valence
electron is S= 1/2, and the orbital angular momentum in this state is L= 0. Thus
the total electron angular momentum is J= 1/2. The first excited state is the P
8
F=I+3/2
F=I+1/2
F=I–1/2
F=I–3/2
2
P3/2
p
F=I+1/2
F=I–1/2
2
P1/2
D1
s
D2
F=I+1/2
F=I–1/2
2
Orbital
Structure
S1/2
Fine
Structure
Hyperfine
Structure
Figure 2.1: Energy level splitting of the ground state and first excited state of an
alkali metal atom. The fine structure splits first excited state further into J=1/2 and
J=3/2 levels. The hyperfine structure further splits the J energy levels. Not drawn
to scale.
shell with L= 1. Due to fine structure L·S coupling the P state splits into 2 P3/2 and
2P
1/2
states. Here we use the standard spectroscopic notation
2S+1 L
J
to describe
the energy levels. Historically the 2 S1/2 → 2P1/2 and 2 S1/2 → 2P3/2 transitions are
referred to as the D1 and D2 lines. See fig. 2.1.
The nuclear spin of the alkali metal I couples to the total electron spin J via the
hyperfine interaction to further split the energy levels into states with good quantum number F. Here F is the total angular momentum F=I+J. It follows from the
Wigner-Eckart theorem (see Cohen-Tannoudji 1972) that the total electron angular
momentum J must be parallel to F. Thus when we probe the hyperfine manifolds
the orientation of the total atomic angular momentum vector we also determine
the total electron angular momentum vector direction.
Implementation of a magnetic field lifts the degeneracy between different Zeeman sublevels of states with the same total angular momentum F, but different
projection m f along the quantization axis defined by the magnetic field. The resulting energy splitting between Zeeman levels is proportional to the field for small
9
field strengths, typically less than a Gauss. This splitting gives rise to Larmor
precession of the atoms between the energy levels with frequency ω L = ∆EL =
γ | B|. Here the gyromagnetic ratio γ for alkali atoms is given by γ ≈ ±2π ×
(2.8MHZ/G)/(2I+1), where I is the nuclear spin of the atomic nucleus and the
sign corresponds to the hyperfine manifold, ie. The F = I + 1/2 manifold yields a
+ sign in the gyromagnetic ratio.
For large magnetic fields however the Zeeman energy level splitting is nonlinear and is given by the Breit-Rabi splitting. We can calculate this by studying
the ground state Hamiltonian of the alkali atom.
The ground state Hamiltonian is of the form:
H = A J I · J + gs µ B S · B − g I µ N I · B
(2.1)
where µ N is the nuclear magneton, A J = 2h̄ωhf /(2I + 1) is the hyperfine coupling
constant, gs ≈2 is the electron g-factor, µ B =9.274×10−24 J/T is the Bohr magneton,
and g I the nuclear g-factor. The hyperfine coupling constant is specific to the atom
under discussion. The energy spectrum can be calculated from the eigenvalues of
the Hamiltonian to be (Corney (1977)):
h̄ωhf
h̄ωhf
E( F = I ± 1/2, m F ) = −
− g I µ N Bm F ±
2(2I + 1)
2
r
x2 +
4xm F
+1
2I + 1
(2.2)
Where
x≡
( gs µ B + g I µ N ) B
2( gs µ B + g I µ N ) B
=
(2I + 1) A J
h̄ωh f
(2.3)
Notice that the energy spacing as a function of the magnetic field is now nonlinear. In the low field adjacent sublevels would be separated by g I µ N Bm F . This
10
clearly scales with the magnetic field. In the high field regime the last term in eq.
2.2 causes the splitting to no longer be directly proportional to the magnetic field.
2.1.2 Optical Pumping
The experiments described in this thesis require a large source of polarized spin.
One could thermally polarize the sample using brute force by introducing the sample into a large magnetic field:
Pther = tanh
1
2 gs µ B B
kB T
!
(2.4)
where gs ≈2 is the electron g-factor and µ B =9.274×10−24 J/T is the Bohr magneton.
The polarization can only be raised to 2% at room temperature if one implements
large magnetic fields on the order of 10T . For comparison the typical absolute
thermal polarization is only 1 × 10−7 at room temperature, in Earth’s field. The
more elegant technique of optical pumping can yield alkali polarization on the
order of unity.
To describe optical pumping we consider a toy model where the atom has no
nuclear spin. However, the following scheme is of a general nature and can be
extended to the case where I6= 0. Typically one utilizes the ground state D1 transition of an alkali metal for optical pumping for a variety of reasons. First, the
alkali doublet have strong oscillator strength which leads to larger absorption by
the pump laser. Second, it is theoritically possible to polarize the ground state to
unity if the D1 is excited (Franzen & Emslie (1957)). In the case of D2 pumping the
maximum achievable polarization is limited to 1/2. This transition is not efficient
for optical pumping. It can be utilized under conditions where the gas density
is rare, and there is little collisional mixing in the excited P state. The details of
collisional mixing will be described shortly.
11
+
σ
g
pin
m
Pu
2
P1/2
2
S1/2
Quenching
Quenching
Collisional Mixing
Spin Relaxation
mJ = +1/2
mJ = -1/2
Figure 2.2: Optical pumping of the electron spin of an alkali.
The experiments in this work rely on optically pumping alkali by tuning to
the D1 transition. To describe this process first consider the D1 2 S1/2 → 2P1/2
transition of an atom in a magnetic field as shown in fig.2.2. Let us define the
quantization axis along the direction of the magnetic field. The ground and excited
state sublevel degeneracy is lifted in the presence of a magnetic field and can each
be resolved into states with magnetic quantum number m J .
The goal of optical pumping is to increase the ground state population of one
of the magnetic sublevels, 2 S1/2 (m J = +1/2) in this example. To accomplish this
we first utilize σ+ photons to excite the 2 S1/2 (m J = −1/2) →2 P1/2 (m J = +1/2)
transition. In the excited state two effects occur. The first is spontaneous emission
to both 2 S1/2 magnetic sublevels, and the second is collisional mixing in the excited
state (Walker & Happer (1997)).
Due to the nonzero orbital angular momentum of the excited state the alkali
wavefunction is not spherically symmetric. This causes collisional mixing effects
to occur. Collisional mixing transfers the atom from the excited P m J = +1/2 to
the m J = −1/2 sublevel. Typically when alkali metal is optically pumped the alkali vapour is in the vicinity of other gases. These can include noble gas, and a
12
molecular quenching gas. The collision between alkali vapour, and noble buffer
gas leads to collisional mixing in the excited state. Physically these may be described as the electron orbital angular momenta coupling to the rotational angular
momenta of the molecule temporarily created by the pair of colliding atoms (Wu
et al. (1985))(Walker & Happer (1997)).
In general the Hamiltonian of such an interaction can be expressed as (Wu et al.
(1985)):
Hcm = γ(r )S · N
(2.5)
Here the S refers to the alkali spin, N to the rotational angular momentum of the
temporarily formed molecule, and γ(r ) as the coupling between the two. This last
factor depends on the inter-atomic potential, the separation of the constituents of
the temporarily formed molecule, and the corresponding wavefunction density of
the pair.
During collision the degree to which the alkali wavefunction is perturbed determines the collisional mixing. Atoms with large polarizability have greater collisional mixing cross-sections. The ground state of the alkali is less susceptible to
wavefunction deformation than the P state sublevels because it is spherically symmetric (Corney (1977)).
If the pumping scheme is iterated the atoms eventually populate the 2 S1/2 m J =
+1/2 state, as atoms in this state can no longer absorb the σ+ light. This represents
an ideal case. There exist mechanisms to prevent this. Fluorescent light from the
spontaneous decay of the excited P state can interact with other alkali in the sample
and excite atoms out of the 2 S1/2 m J = +1/2 sublevel. This is referred to in the
literature as radiation trapping. This can be prevented by introducing a molecular
quenching gas. A typically quenching agent is N2 . N2 molecules collide with the
excited alkali atoms and transfer them to the ground state without re-irradiation
of the alkali (Happer (1972)). This can occur because the N2 molecule has a large
13
number of closely spaced energy levels, due to its rotational, and vibrational level
structure. As the N2 molecules return to their ground state they may either transfer
the energy from the alkali atom to its rotational or vibrational modes to be reradiated at a frequency different from the D1 line. The excited alkali atoms become
non-radiatively quenched. Typically 100 Torr of Nitrogen is sufficient to prevent
radiation trapping in cells with alkali densities near 1014 /cm3 . There are also spinrelaxation mechanisms which can transfer atoms between the two ground state
magnetic sublevels to depolarize the sample. These will be discussed in the next
section.
The alkali nucleus is strongly coupled to the total electron angular momentum via the hyperfine interaction. Thus when we polarize the electrons we also
successfully polarize the alkali nuclei. This leads to some interesting effects. The
alkali electron spin polarization decay is slower than one would initially estimate.
This is due to the electron spins being re-polarized via the hyperfine interaction
with the polarized nucleus. This effect is termed the slowing down factor, and will
be quantitatively described later in this work.
2.1.3 Effects of Pumping Rate and resonance Lineshape on Optical Pumping
In describing optical pumping it is useful to define a quantity named the optical
pumping rate. The optical pumping rate is defined as the rate at which an unpolarized alkali atom absorbs photons from the pump laser.
Rop =
Z
I (ν)
σ (ν)dν
hν
(2.6)
I (ν) is the light spectral density in units of Watts cm−2 Hz−1 , and σ (ν) is the photon
absorption cross section.
14
The photon absorption cross section is defined by the atomic response as a function of incident photon frequency. In general it is influenced by the natural lifetime
of the atomic state of interest, pressure broadening of the atomic resonance, and
Doppler broadening of the resonance due to thermal velocity distribution. The
absorption cross section can be related to the classical electron radius by:
Z ∞
0
σ (ν)dν = πre c f
(2.7)
This is valid regardless of the spectral line shape. In eq.2.7 the oscillator strength
f is the quantum mechanical correction factor to the classical expression for the
relation between absorption cross section, and classical electron radius. It is a dimensionless number and depends on which transition we are pumping. For the
D1 line it is ≈ 1/3, and for D2 it is ≈ 2/3(Migdalek & Kim (1998)). The absorption
cross section is given by:
σ (ν) = πre c f Re[V(ν − ν0 )]
(2.8)
where V(ν − ν0 ) is the atomic lineshape. It is given by the Voight profile (Happer
& Mathur (1967)), and includes the effects of pressure broadening, natural lifetime,
and Doppler broadening.
V(ν − ν0 ) =
Where
Z ∞
0
′
′
L(ν − ν )G(ν − ν0 )dν
′
!
√
√
2 ln 2[(ν − ν0 ) + iΓ L /2]
2 ln 2/π
w
V(ν − ν0 ) =
ΓG
ΓG
(2.9)
(2.10)
And the function w is given in terms of the complex error function as:
2
w( x ) = e− x (1 − erf(−ix ))
15
(2.11)
Gas
He
He
Ne
Ne
Ar
Ar
Kr
Kr
Xe
Xe
N2
N2
Energy Level
P1/2
P3/2
P1/2
P3/2
P1/2
P3/2
P1/2
P3/2
P1/2
P3/2
P1/2
P3/2
Half-halfwidth
1.55 ± 0.03
2.06 ± 0.04
0.85 ± 0.02
1.16 ± 0.04
2.45 ± 0.03
1.98 ± 0.03
2.31 ± 0.05
2.31 ± 0.05
2.75 ± 0.03
2.75 ± 0.03
2.45 ± 0.03
2.45 ± 0.03
Shift
+0.45 ± 0.04
+0.24 ± 0.04
−0.41 ± 0.02
−0.62 ± 0.02
−2.31 ± 0.05
−1.52 ± 0.04
−1.65 ± 0.04
−1.16 ± 0.04
−1.79 ± 0.06
−1.79 ± 0.06
−1.83 ± 0.04
−1.32 ± 0.04
Table 2.1: Comparison of the line broadening and shift parameters for K in various
Gases. All experimental measurements were made in the 400 − 420K temperature
regime. The line broadening parameter and shift parameter have units of 10−9 rad
s−1 atom −1 cm3 . Data taken from Lwin & McCartan (1978)
er f denotes the standard error function. Here ΓG is the linewidth of the Gaussian
contribution to the lineshape due to Doppler broadening for an atom of mass M
!
√
−4 ln 2(ν − ν0 )2
2 ln 2/π
exp
G(ν − ν0 ) =
ΓG
Γ2G
(2.12)
r
(2.13)
ν0
ΓG = 2
c
2k B T
ln 2
M
And Γ L is the linewidth of the Lorentzian contribution to the Voight profile due to
pressure broadening.
L(ν − ν0 ) =
Γ L /2π
(ν − ν0 )2 + (Γ L /2)2
(2.14)
The values for Γ L as a function of pressure is listed for K vapour in table 2.1.
16
2.1.4 Evolution of Alkali polarization due to Optical Pumping
The evolution of the polarization of the alkali atoms due to optical pumping can
calculated. In order to compute the polarization achieved via optical pumping one
must clarify whether the target cell contains any buffer gas. For certain applications it becomes advantageous to operate without a buffer gas (see Knappe et al.
(2006),Cates et al. (1988),and Pustelny et al. (2006) for more information), and utilize a cell coating instead. However those applications will not be discussed in this
work.
Once the D1 2 S1/2 → 2P1/2 transition is excited the electron will decay back
to both the m J = 1/2, or m J = −1/2 sublevels. For the case of no buffer gas the
branching ratios are determined by the Clebsch-Gordon coefficients to be 1/3, and
2/3 respectively (Budker et al. (2004)). However when buffer gas is introduced in
the cell, collisional mixing in the excited states alters the branching ratio to be 1/2
for both cases.
Under these conditions we can now calculate the time evolution of the alkali
polarization. Let us denote the populations in the m J = +1/2 and m J = −1/2
sublevels as N+ , and N− respectively. The pumping rate is defined as the rate at
which an unpolarized atom absorbs a photon. The m J = +1/2 state is unable to
absorb a photon if one uses σ+ light. Therefore the rate at which the m J = −1/2
state absorbs a photon is at twice the optical pumping rate. This is because the
optical pumping rate is defined as per unpolarized atom. Using this and including
the 1/2 branching ratio we can describe the pumping process by realizing
−
− dN
dt . Thus:
1
R
R
dN+
= (2Rop ) N− + rel N− − rel N+
dt
2
2
2
dN+
dt
=
(2.15)
Where Rrel is the relaxation rate. By noting that the polarization can be expressed
17
as:
P = 2 < Sz >=
N+ − N−
N+ + N−
(2.16)
the time evolution of the polarization can be ascertained. Assuming P(0) = 0:
P(t) = Pequil (1 − e( Rop + Rrel )t )
(2.17)
During equilibrium:
Pequil =
Rop
Rop + Rrel
(2.18)
This is only strictly true in the case of an atom with zero nuclear spin. The electron
spin and nuclear spin are coupled via the hyperfine interaction. As the electron
spin precesses it drags the nuclear spin along with it. This results in a slower
precession frequency than that of a free electron in a magnetic field (Budker et al.
(2004)). It is modified as:
γ=
γe
2I + 1
(2.19)
Where γe = gs µb /h̄ is the gyromagnetic ratio for a free electron, and has magnitude 2π × 2.8MHz/G. The coupling between the valence electron and the nucleus
also alters the rate at which the valence electron is depolarized. Just as the hyperfine interaction can polarize the nucleus when we pump the valence electron,
the opposite can occur. When the electron spin is depolarized, the nuclear spin
can re-polarize the electron. Thus the actual rate at which the electron is either
depolarized by collisions, or polarized by a laser must be modified. To good approximation one can describe this effect by a linear slowing down factor ǫ (Walker
& Happer (1997)). For atoms in low magnetic field where γB << Rse the system
can be described accurately in terms of a two level system. The equation of motion
18
of the atom can now be described as:
1
1
dS
=
sb
z − S − Rrel S
γe B × S + Rop
dt
ǫ+1
2
(2.20)
This is often referred to as the Bloch equation in the literature. Here the first term
describes the precession of the spins in a magnetic field. The second term describes
the pumping of the spins to their equilibrium value. Here ǫ is given by:
ǫ( I, β) = (2I + 1)coth( β/2)coth( β[ I + 1/2]) − coth2 ( β/2)
(2.21)
(2.22)
P = tanh( β/2)
(2.23)
β = ln
1+P
1+P
Here I is the nuclear spin, and β is the spin temperature of the system (Walker
& Happer (1997)), which can be related to the polarization P. A more complete
discussion on spin temperature will be described later in this work. But for now
it is sufficient to note the special limiting cases for ǫ for low and high polarization
respectively as:
ǫ( I, β << 1) = 4I ( I + 1)/3
(2.24)
ǫ( I, β >> 1) = 2I
(2.25)
Although this alters the time evolution of the electron spin polarization to:
P(t) = Pequil (1 − e( Rop + Rrel )t/(ǫ+1) )
The equilibrium polarization reached remains the same eq 2.18.
19
(2.26)
2.1.5 Dynamics of polarized alkali at low magnetic fields
Optical pumping is used to polarize the spins of an alkali species. An application of
the polarized alkali spin of particular importance is the determination of magnetic
fields. This is accomplished by determination of the precession frequency of the
alkali by monitoring the spin dynamics and relating it to the magnetic field.
The alkali undergo Larmor precession due to the interaction of the spins with
the magnetic field:
H = γalkali h̄~B · ~S
(2.27)
The dynamics of the alkali due to this interaction gives a response:
i
d~
S = [H, ~S]
dt
h̄
(2.28)
Noting that the components of the spin follow the commutation relation:
[Sx , Sy ] = iSz
(2.29)
we can describe the evolution of Sz as
d~
Sz = iγalkali (−iBx Sy + iBy Sx )
dt
(2.30)
One can describe the evolution of the Sx , and Sy components of the spin with similar equations. Inclusion of the effects of optical pumping, and spin relaxation
modify the spin evolution given by eq.2.30 to that given by the phenomonological
equation eq.2.20.
The magnetometer bandwidth can be calculated utilizing a set of simplified
Bloch equations. Assume we impose an oscillating field of the form By = B0 exp(−iωt).
Let us rewrite eq.2.20 in terms of the components of the alkali polarization Pex , and
20
Pze . Then, for a magnetometer where Bx , and Bz are set to zero imposing a field By
gives the following behaviour:
1
dPxe
= γe By Pze − Rtot Pxe
dt
ǫ+1
1
dPze
= −γe By Pxe − Rtot Pze + Rop
dt
ǫ+1
(2.31)
(2.32)
The first term represents precession about By , the second represents spin relaxation, and the third term represents optical pumping. Solution of the above equations yields:
Pxe =
Pze γe B0
Rtot − iω (ǫ + 1)
S = R( Pxe ) =
Pze γe B0
R2tot − ω 2 (ǫ + 1)2
(2.33)
(2.34)
ω0 is the frequency of the spin precession in the ambient magnetic field. For low
frequencies the signal can be simplified to obtain:
S=
2.2
Pze γe B0
Rtot
(2.35)
Spin Relaxation
In order to achieve optimal sensitivity and minimize the magnetic linewidth during optical pumping experiments, one must maximize the spin polarization lifetime. Since magnetic linewidth is correlated to the lifetime of the Zeeman states,
we are effectively maximizing the spin polarization. In general we must consider
two cases, the longitudinal polarization lifetime T1 , and the transverse polarization lifetime T2 . Here T1 is the lifetime of the polarization parallel to the magnetic
holding field, and T2 is the lifetime of polarization in a perpendicular direction.
21
One can express the longitudinal lifetime T1 as:
1
1
=
( R + Rop + R pr ) + Rwall + Rinh
T1
ǫ + 1 sd
(2.36)
Here the first term is due to spin destruction collisions. These collision may be
of several types, including collisions between alkali atoms, collisions between the
alkali atoms and the buffer gas, or collisions between the alkali atoms and the
quenching gas.
sel f
bu f f er
Rsd = Rsd + Rsd
quench
+ Rsd
(2.37)
Each of the collisional spin destruction rate mechanisms can be described by:
Rsd = nσv
(2.38)
where n is the density of the gas species in question which is colliding with the alkali, and σ is the spin destruction cross section. Calculation of the spin destruction
crpss-sections can be performed from a knowledge of the atomic wavefunction,
interaction potential and the interaction Hamiltonian. These calculations are normally accurate to 50% with the measured value, due to imprecise knowledge of
the specifics of the collisional interaction (Walker & Happer (1997)). The last term
v is the relative velocity of the colliding pair and is given by:
v=
r
8κ B T
πM
(2.39)
and M is the reduced mass of the alkali and its colliding partner:
1
1
1
= + ′
M
m m
(2.40)
The second term in eq (2.36) is the optical pumping rate. This affects T1 be22
cause the absorption of the pump beam alters the angular momentum state of the
alkali. The next term is similar, but due to absorption of the probe beam. These
relaxation mechanisms depolarize the valence electrons, while leaving the nucleus
unaffected. Thus the sum of the first three terms must be divided by the slowing down factor because of the previously mentioned re-pumping of the electron
due to the polarized nucleus. The next term is due to collisions with the wall. In
uncoated cells collisions with the cell wall completely depolarize the alkali atoms.
The last term is due to magnetic field inhomogeneity. The pumped alkali align
with the net magnetic holding field. If a gradient is present it can locally alter the
direction of the net magnetic field and change T1 .
Consider the mean free path λ in the gas. Since the Larmor frequency is much
faster than the transit rate v/λ between atomic collisions the atomic polarization
follows the total magnetic field. This leads to relaxation when the atom changes
direction after a collision (Budker et al. (2004)). The atom experiences a small magnetic field, which varies slowly compared to the Larmor frequency. It is transverse
to the holding field and corresponds to a rotation of the total magnetic field vector
with a frequency ω ≈
δBv
Bholding λ .
It can be treated as the flipping probability of an
atom with spin oriented along a magnetic holding field, by the presence of a fluctuating transverse magnetic field. Magnetic field gradients are normally not the
dominant alkali relaxation mechanism. Typical gradients found in the low field comagnetometer experiment are 10µG/cm, with a holding field of 1mG. This gives a
negligible alkali relaxation rate due to gradient relaxation.
Additionally magnetic field gradients cause the atoms to precess due to local
variation in the magnetic field and de-phase. This causes relaxation of the transverse component of the spins. Additionally the mechanisms discussed earlier pertaining to the T1 relaxation similarly affect the transverse polarization because they
23
also randomly orient the spin polarization. The T2 lifetime can thus be given as:
1
1
1
+
Rse + R grad
=
T2
T1 qse
At high field where γB >> Rse Happer & Tam (1977) give
(2.41)
1
qse
as:
1
2I (2I − 1)
=
qse
3(2I + 1)2
Whereas at low magnetic field where γB << Rse they show that
(2.42)
1
qse
→ 0. Rse is
the spin exchange rate between alkali atoms, and Rgrad is the dephasing due to
nonuniform magnetic field across the sample cell. The significance of eq. (2.42)
and its physical description will be described in more detail in the next section.
In order to maximize the sensitivity of most polarization experiments one attempts to maximize the T2 because it is the relevant parameter in the determination
of the precession frequency. One also attempts to maximize T1 in a sense because
it limits the maximum achievable value of T2 . It would be prudent to review each
of the aforementioned pumping and relaxation mechanisms in more detail. In the
next section we do so.
2.2.1 Spin Exchange Collisions
When alkali atoms collide there is the possibility of their exchanging spin states
(Purcell & Field (1956)). This can be shown as:
|+i A |−i B → |−i A |+i B
(2.43)
At large alkali density this has been shown to be the dominant relaxation mechanism. The following treatment on spin exchange collisions follows the treatment
by Budker et al. (2004).
24
One can describe the spin exchange mechanism by realizing that the interatomic potential during collision has a spin dependent contribution:
V (r ) = V0 (r ) + S A · SB V1 (r )
(2.44)
Where V0 is a spin independent interaction term, and S A and SB are the respective
spins of the colliding alkali. The wavefunction of a free atom before collision can
be expressed in the |S, MS i basis in terms of the singlet |0, 0i, and triplet |1, 0i states
as:
1
|ψ(0)i = |+i A |−i B = √ (|1, 0i + |0, 0i)
2
(2.45)
During collision the singlet and triplet states acquire a relative phase
1
|ψ(0)i = √ (|1, 0i + ei∆φ(t) |0, 0i)
2
(2.46)
where the relative phase acquired is:
2π
∆φ(t) =
h
Z t
0
V1 [r (t)]dt
(2.47)
The atoms have a probability of undergoing a spin exchange collision when ∆Φ is
an odd multiple of π. In this case
|ψ(t)i → |−i A |+i B
(2.48)
Though spin exchange collisions conserve total mF quantum number during
collision they can alter the F state of the alkali. These collisions are fast with respect to the nuclear hyperfine interaction, thus they do not affect the nuclear spin
state. These collisions also cause decoherence of the transverse spins, and dephasing because the two hyperfine manifold actually have gyromagnetic ratios which
25
F=2
-2
+ωL
F=1
-1
-1
+ωL
−ωL
0
0
+ωL
−ωL
+1
+ωL
+2
+1
Figure 2.3: Ground-state Zeeman sublevels for the case I=3/2. Sublevels are labeled by their m F azimuthal quantum number. Note that the energy level splitting
changes sign for the different F manifolds. This causes both F manifold to have
gyromagnetic ratios with opposite sign
Figure 2.4: During spin-exchange collisions the total angular momentum F 1 + F 2 is
conserved, the atoms may switch between m F hyperfine sublevels. The hyperfine
levels F = I ± 1/2 are represented here by the colors red and blue. The atoms
is different hyperfine levels have gyromagnetic ratios with different sign,t equal
magnitude. They will precess in opposite directions and decohere.
while having the same magnitude differ in sign. See fig.2.3 and fig.2.4.
This can readily be seen by solving the ground state alkali Hamiltonian eq.
(2.1)(Walker & Happer (1997)): and solving for the eigenvalues in the | F, m F i basis.
ω F= I +1/2 = −ω F= I −1/2 = 2π
gs µ B
(2I + 1)h
(2.49)
This decoherence causes a broadening of the Zeeman sublevels which at high
26
field and low polarization can be expressed as (Happer & Tam (1977)):
2I (2I − 1)
1
=
qse
2(2I + 1)2
(2.50)
This expression is accurate in the high field regime where ω Larmor >> Rse . This
effect also contributes to the reduction of the T2 of the alkali atom precession. See
eq. (2.41).
In the low field limit where ω Larmor << Rse we find interesting behaviour as
spin exchange processes no longer contribute to the transverse decoherence of the
spins. This can be explained if we think in terms of the relative populations of the
different hyperfine states. When the spin exchange rate is much higher than both
the optical pumping and relaxation rates the atoms mix the m f sublevel populations. This condition applies at high alkali density when spin exchange collisions
occur frequently. Here the steady state m f sublevel population can be described
by a spin temperature distribution (Anderson et al. (1959),Anderson et al. (1960)).
See fig.2.5.
ρ( F, m F ) =
1 βm F
e
ZF
(2.51)
where
ZF = Σe βm F =
sinh[ β( F + 1/2)]
sinh( β/2
(2.52)
Here the sublevel population possess a Boltzmann like distribution but with a spin
temperature β instead of the normal kinetic thermal temperature. We find that
because of the high spin exchange rate the atoms spend time in both hyperfine
manifolds. However they do so with a probability corresponding to the spin temperature distribution. Since the spin exchange rate is much higher than the Larmor
precession rate the atoms motion can be thought of as a slower averaged precession due to the fact that the gyromagnetic ratios in each hyperfine manifold has
opposite sign but have unequal time spent in each manifold. This results in a net
27
F=2
e-2β
F=1
mF=-2
e-β
1
e+β
e-β
1
e+β
mF=-1
mF=0
mF=+1
e+2β
mF=+2
Figure 2.5: When the rate of spin-exchange collisions is high, the Zeeman sublevel
populations are given by a Boltzmann distribution which is characterized by a
spin-temperature. The sublevel population in this case scales as eβm F . The case of
nuclear spin I= 3/2 is shown.
rotation of the spins in one direction.
The alkali experience spin exchange collisions and hop between the two hyperfine manifolds. The two hyperfine manifolds possess gyromagnetic ratios with
opposite sign. The atoms precess a small amount between collisions. However
the atoms spend a larger amount of time in the hyperfine manifold with larger F
value. This is determined by the spin temperature distribution. Thus even though
the atoms hops back and forth between the hyperfine manifolds it preferentially
spends a larger fraction of time in the hyperfine manifold with the larger F value.
This causes the atom to have a net precession in the orientation consistent with the
sign of the gyromagnetic ratio of the larger F valued hyperfine manifold. Since
this precession is coherent, spin exchange is effectively removed as a source of relaxation to first order. The spin exchange relaxation rate becomes (Allred et al.
(2002)):
Rse =
( Q)
2
ωser
f Tse
2
− (2I + 1)2
2
(2.53)
Q is described in eq. (2.55). In the equation above Tse is the time between spin exchange collisions, and depends on the specifics of the cell temperature and buffer
28
gas pressure. However under typical conditions of the experiments in this thesis
it is ≈ 15µS. Optical pumping schemes operating in this regime are called spin
exchange relaxation free (SERF). The rate of precession is modified from the traditional Larmor rate by (Allred et al. (2002)):
ωSERF =
2πgs µ B B
( Q)h
(2.54)
Q = 1+
I ( I + 1)
S ( S + 1)
(2.55)
Where
for high polarization. One finds the dependence of ǫ on polarization as (Savukov
& Romalis (2005)):
Q( P) =
2I + 1
2 − 32I++P12
(2.56)
2.2.2 Spin Destruction Collisions
Spin destruction collisions are the next leading mechanism for spin relaxation. Pictorially they may be represented as:
|↑i A + |↓i B → |↓i A + |↓i B
(2.57)
They occur through when the spin angular momentum becomes coupled to the
orbital angular momentum of the atom-atom collision. This may be between the
alkali atoms and any of the other alkali atoms, buffer gas atoms, or quenching gas
molecules. This angular momentum coupling is hypothesized to be due to a spin
axis interaction (Bhaskar et al. (1980)):
VSA =
2
bR
b − 1) · S
λS · (3 R
3
29
(2.58)
Alkali Metal
K
Rb
Cs
σsel f
1 × 10−18 cm2
9 × 10−18 cm2
2 × 10−16 cm2
σHe
8 × 10−25 cm2
9 × 10−24 cm2
3 × 10−23 cm2
σNe
1 × 10−23 cm2
−−
−−
σN2
−−
1 × 10−22 cm2
6 × 10−22 cm2
Table 2.2: Spin destruction cross sections of alkali atoms with various gases.
Adapted from Allred et al. (2002)
A list of spin destruction cross sections of alkali gases with typical buffer, and
quenching gases is listed in table 2.2.
2.2.3 Diffusion wall collisions, and Magnetic field Gradients
When alkali interact with the cell walls they normally become completely depolarized. Physically this process involves the alkali becoming adsorbed on the glass
surface. Though the adsorption time is typically between 10µ S and 100nS, the
atoms experience very large magnetic and electric field emanating from the glass
surface during this time. This is sufficient to depolarize both the spin of the electron and the nucleus. In order to minimize this effect one can either treat the cell
wall with an anti-relaxation coating or fill the cell with a high pressure of buffer
gas to decrease the alkali diffusion to the cell wall. In this work we consider the
second option.
If one assumes that the alkali becomes fully depolarized when it becomes adsorbed on the wall then one can model the polarization in the cell by the diffusion
equation.
d
P = D ∇2 P
dt
(2.59)
Here D is the diffusion constant, where λ is the mean free path, and v the average
thermal velocity.
D=
1
λv
3
30
(2.60)
The diffusion constant for K in neon is given by(Franz & Volk (1982)):
√
DK − Ne = 0.19cm2 /s
1 + T/273.15K
p amagat
!
(2.61)
Where the rate of K relaxation for the case of a spherical cell is found by solving
eq. (2.59):
Rdi f f = DK − Ne
π 2
(2.62)
a
Here the temperature T is given in Celsius, and p is the buffer gas pressure in
amagat, and a is the cell radius. This is only strictly valid for a spherical cell.
Franzen (1959) gives the diffusion constant of Rb in Ne as:
√
DRb− Ne = 0.31cm2 /s
1 + T/273.15K
p amagat
!
(2.63)
The alkali may also experience relaxation due to traveling through a magnetic
field gradient. The specifics of this depend on cell geometry, diffusion constant,
and magnetic gradient orientation. The gradient relaxation rate is defined as:
R grad = D
∇B
B
2
(2.64)
Where δB is the magnetic field variation over the cell, and D is the diffusion constant.
The variation in the magnetic field also causes broadening of the alkali precession frequency ≈ γ∇ B. This local variation of the field also causes dephasing of
the spin precession and places a limit on T2 , the transverse relaxation time. For a
formal treatment of this effect see Cates et al. (1988). We take a more simplified
approach to describe this effect.
We can compute this by imagining the cell partitioned into two halves each
differing in field by δB. In the regime where the atoms are free to move about the
31
cell unimpeded they dephase according to:
2γδBT2 ≈ 1
(2.65)
However in actuality the atoms travel across the cell and experience a δB given by
the time averaged field. This movement can be treated as a random walk across
the cell. Thus the difference in averaged field experienced by two alkali atoms is:
δB
δBavg ≈ √
N
(2.66)
Where N is the number of collisions before dephasing. N can be expressed in terms
of T2 as
N≈
v
T2
R
(2.67)
Here R is the cell dimension. Noting that one can generalize this argument from a
cell of dimension R to a buffer gas included cell of mean free path λ one can solve
this set of equations to find:
λ
1
> (2γδB)2
T2
v
2.3
(2.68)
Monitoring polarized Alkali
2.3.1 Optical Rotation
In order to determine the orientation of the alkali polarization we utilize the technique of optical rotation. A linearly polarized beam, which is slightly detuned
from the optical transition frequency utilized for pumping, is employed as a probe
beam. One uses a weak probe beam so that the alkali vapour is not significantly
pumped along the probe direction. As the probe beam interacts with the magnetized sample in the ground state the plane of rotation on the probe beam rotates.
32
θ
Figure 2.6: When linearly polarized light passes through a medium which is polarized the axis of polarization of the light rotates. This rotation angle is proportional
to the projection of atomic spin along the propagation direction.
See fig.2.6. To understand this recall that the linear polarized beam is equivalent
to a composition of two equal beams which are circularly polarized with opposite
helicity. The rotation occurs when the index of refraction of the probing transition
differs for the transitions of different helicity.
To describe this effect we follow the approach of Mort et al. (1965), and Erickson (2000). To describe optical rotation mathematically let us first define the probe
beam in terms of electromagnetic waves. The electric field of such waves propagating along the xb direction can be described as:
E (0) =
E (0) =
E0
yb + c.c.
2
E0
E0
(yb + ib
z) + (yb − ib
z) + c.c.
4
4
(2.69)
(2.70)
Where c.c is the complex conjugate. After propagating the length of the cell l =
tc/n(ν) the electric field along the yb becomes:
E(l ) =
E0 iωn+ (ν)l/c
E0
e
(yb + ib
z) + eiωn− (ν)l/c (yb − ib
z) + c.c.
4
4
(2.71)
Where n+ , and n− are the indices of refraction for σ+ and σ− polarized light re-
33
spectively. It becomes convenient to define the quantities:
n(ν) = [n+ (ν) + n− (ν)]/2
(2.72)
∆n(ν) = [n+ (ν) − n− (ν)]/2
(2.73)
We can then substitute this into eq. (2.71) to obtain:
E(l ) =
E0 iωn(ν)l/c iω∆n(ν)l/c
E0
e
e
(yb + ib
z) + eiωn(ν)l/c e−iω∆n(ν)l/c (yb − ib
z) + c.c. (2.74)
4
4
One can ignore the common phase factor exp(iωn(ν)l/c when determining the
optical rotation angle. The rotation angle is defined to be:
θ=
πνl
Re[n+ (ν) − n− (ν)]
c
(2.75)
We can substitute this into eq. (2.74) to find the electric field vector of the emerging
probe beam as:
E(l ) = E0 (cosθ yb − sinθb
z)
(2.76)
One can show that an atomic vapour is a medium where through appropriate
choice of laser frequency n+ (ν) 6= n− (ν). To validate this assertion let us consider the following. First the index of refraction of light for the D1 transition is
given by:
n(ν) = 1 +
nre c2 f
Im[V(ν − ν0 )]
4ν
(2.77)
Where f is the familiar oscillator strength, and V(ν − ν0 ) is the Voight profile of the
transition. We notice that when the probe beam interacts with the ground state,
only transitions which obey the quantum selection rules ∆m j = ±1 will occur
depending on the helicity of the light. Each σ± component of the linearly polarized
probe beam then couples to the ground state depending on the population of the
34
D1 Transition
D2 Transition
2
2
P3/2
1
1
3/4
2
mJ = -1/2
P3/2
mJ = +1/2
1/4
1/4
3/4
mJ = +1/2
mJ = +3/2
2
S1/2
S1/2
mJ = -3/2
mJ = -1/2
Figure 2.7: Branching ratios for the D1 and D2 transitions.
ground state sublevel. Thus we can effectively write:
nre c2 f D1
Im[V(ν − νD1 )]
4ν
(2.78)
nre c2 f D1
Im[V(ν − νD1 )]
n− (ν) = 1 + 2ρ(−1/2)
4ν
(2.79)
n+ (ν) = 1 + 2ρ(+1/2)
For an unpolarized vapour ρ(+1/2) = ρ(−1/2). However for a polarized vapour,
Px =
ρ(+1/2)−ρ(−1/2)
,
ρ(+1/2)+ρ(−1/2)
we see that n+ 6= n− . Substituting this into equation 2.75,
and recalling that ρ(+1/2) + ρ(−1/2) = 1 we find:
θ=
πlnre cPx
(− f D1 Im[V(ν − νD1 )])
2
(2.80)
One can generalize this result taking into account the effect of the D2 line on the
probe beam to obtain:
πlnre cPx
θ=
2
1
− f D1 Im[V(ν − νD1 )] + Im[V(ν − νD2 )]
2
(2.81)
The negative sign and factor of 1/2 can be attributed to the different branching
ratios for the D2 transition. See fig.2.7.
To detect the rotation of the polarization angle one uses a polarizing beam splitter to split the probe beam into two orthogonally polarized beams. These are fed
into two photo-diodes whose output is fed through a subtracting photodiode am35
Polarizing
Beamsplitter
Linear
Polarizer
Cell
Probe Laser
Photodiode
Photodiode
Figure 2.8: Optical detection with a linearly polarized probe beam passing through
a polarized cell, then being split by a polarizing beam splitter into two separate photodiode detectors. The resultant signals from each photo-detector is fed
through a photodiode amplifier and then into a subtraction circuit, giving the rotation.
plifier. See fig.2.8.
The plane of polarization of the probe beam is normally adjusted so that one
obtains equal signal in each photodiode in the case where there are no polarized
alkali. This corresponds to the point where the beam splitter is at 45◦ to the probe
beam polarization. Then the difference signal is directly proportional to the polarization. In this case the signal on each photodiode is given by:
I1 = I0 sin2 (θ −
π
)
4
(2.82)
I2 = I0 cos2 (θ −
π
)
4
(2.83)
where I0 is the intensity of the probe beam, and I1 and I2 are the intensities on the
two photo-detectors. Solving for the rotation angle in radians we get:
1
Θ = sin−1
2
I1 − I2
I1 + I2
(2.84)
Which is often useful to write as:
6
20
2
Θ = Φ 1 + Φ3 + Φ5 + Φ7 + · · · · · · · · ·
3
5
7
36
(2.85)
Linear
Polarizer
Photodiode
Input
Faraday
Modulator
Cell
Lock-In
Amplifier
Linear
Polarizer
Probe Laser
Reference
Figure 2.9: Optical detection with a linearly polarized probe beam passing through
a Faraday rotator, then a polarized cell, a second linear polarizer at 90degrees to the
first, and finally a photodiode detector. The photo-detector signal is fed through
a photodiode amplifier and using a Lockin amplifier referenced to the Faraday
modulator frequency. The in phase component of the Lockin amplifier gives the
rotation
where
Φ=
I1 − I2
2( I1 + I2 )
For the case where Φ << 1 we have Θ =
I1 − I2
.
2( I1 + I2 )
(2.86)
For sensitive polarimetry one
often alters the detection scheme to increase the sensitivity or decrease the 1/ f
noise of this measurement.
An example of this is to send the probe beam through a Faraday modulator
which modulates the plane of polarization of the probe beam by a small angle via
application of strong internal magnetic fields. See fig.2.9. In this arrangement one
first passes the probe beam through a plane polarizer, then through the Faraday
modulator, then the cell, and finally through a second polarizer set at 90◦ to the
first polarizer before finally terminating on a photodiode. The resultant signal on
the photodiode is:
I = I0 sin2 [θ + αsin(ωmod t)]
(2.87)
Here α denotes the amplitude of modulation of the probe polarization axis, I0 denotes the light intensity transmitted through the cell and ωmod denotes the frequency of modulation of the Faraday modulator. To low order one can Taylor
37
expand the eq. (2.87) to obtain:
I ≈ I0 [θ 2 + 2θαsin(ωmod t) + α2 sin2 (ωmod t)]
(2.88)
One then references the detected signal to a lockin amplifier to detect only the
Fourier component of the signal at the Faraday modulation frequency. This method
greatly reduces 1/ f noise and gives a signal:
ILockin ≈ 2I0 θα
(2.89)
Another commonly used detection scheme is to employ a dichroic plate before
the polarizing beam splitter in the standard detection arrangement. The dichroic
plate is a type of poor polarizer which greatly attenuates but does not extinguish
one particular orientation of polarized light. These often have extinguish or attenuation ratios on the order of 100 : 1. The attenuating axis of the dichroic plate is
aligned with the polarization axis of the probe beam. This increases the magnitude
of the signal in one photo-detector relative to the other, and effectively increase the
rotation angle. This however does not alter the maximum sensitivity as the relative noise on both channels remains unaffected. This system also requires a careful
calibration by comparing the output signal as the probe polarization axis is varied
because the signal to angle conversions factor is a function of angle.
2.3.2 Light Shifts
When operating with an off resonant, or nearly resonant laser beam one must take
special precautions to avoid the effects of light shifts. Light shifts mimic the behaviour of a magnetic field and will alter both the precession frequency, and orientation of free spins. Light shifts can arise from two different mechanisms. Our
38
description will follow that by Appelt et al. (1998).
The mechanism for light shifts typically encountered in precision frequency
measurements is due to the AC Stark effect of the electric field of the light beam
interacting with the atomic vapour. This interaction can be described by:
δH = ∆Eν −
ih̄
< Γ >= − E∗ · α(ν) E
2
(2.90)
Here the energy shift is given by ∆Eν , α(ν) describes the complex atomic polarizability, and < Γ > is the average photon absorption rate. It can be related to the
photon flux Φ, and the absorption cross-section σ (ν) by:
< Γ >= σ(ν)Φ(1 − s · S)
(2.91)
where S is the atomic spin, and s is the photon spin vector. This is given by:
s = ib
ǫ×b
ǫ∗
(2.92)
where b
ǫ is the unit Jones polarization vector. Jones vectors are a convenient way to
describe polarization. In Jones terminology a σ+ , σ− , and π polarized beams are
given by:
ǫσ +
ǫσ −


 1 

1 
=√ 
i 

2 

0


 1 

1 
=√ 
−i 


2

0
39
(2.93)
(2.94)


 1 
 

ǫπ = 
 0 
 
0
(2.95)
Because the Hamiltonian is an analytic function, its real and imaginary parts are
related. One can apply the Kramers-Kronig relations to eq. (2.90) to find the energy
shift to be:
∆Eν =
h̄
πre c f Φ(1 − 2s · S) Im[V(ν − ν0 )]
2
(2.96)
One can readily see that this has the same form as the Zeeman interaction:
∆Eν = h̄γe B LightShi f t · S
(2.97)
where the effective magnetic field the atom experiences is:
B LightShi f t =
−πre c f Φ
Im[V(ν − ν0 )]s
γe
(2.98)
Here we ignore the common offset given by eq. (2.96) and only retain the relevant
spin dependent energy shift. One can generalize this and include the effect of the
D2 line by replacing s in the above equations by s/2.
Since the light shift interaction appears as a fictitious magnetic field the atoms
respond by precessing around the total effective magnetic field B = B0 + B LightShi f t .
This effect can be quite large when B0 ≈ B LightShi f t . Thus we try to minimize
this effect whenever possible. Note that the eq. (2.98) becomes zero either when
Im[V(ν − ν0 )] or s become zero. The first case corresponds to a laser beam exactly
on resonance, as Im[V(ν − ν0 )] has a dispersive shape. One can easily verify using
the Jones vectors that the second case where s vanishes corresponds to the case of
linearly polarized light. To minimize the effects of light shifts we ensure that the
40
pump beam is tuned to resonance. We must also ensure that beams which are off
resonance, such as the probe, are linearly polarized.
There exists a second mechanism by which light shifts can occur in the optical
pumping system. During optical pumping atoms spend a small amount of time
out of the ground state and in the excited state. The gyromagnetic ratio of the
alkali is different in the excited state. Thus atoms which were not pumped into the
excited state will acquire a phase relative to atoms which remained in the ground
state the entire time. This effect is generally much smaller than the precession
frequency itself and is negligible in our systems.
2.4
Coupled Spin Dynamics
In a vast number of fundamental physics experiments such as electric dipole moment search (Romalis et al. (2001)), CPT violation (Kornack et al. (2008)), and pulmonary imaging (Oros & Shah (2004)), it would be useful to polarize noble gas
nuclei. Due to spin dependent interactions the electron spin of the alkali can be
used to polarize the nuclear spin of a noble gas. This is called spin exchange optical pumping (SEOP). This is the typical way in which noble gases are polarized,
with the exception of helium. In some cases it is advantageous to polarize helium
gas via a technique called metastable exchange optical pumping. That technique
will not be discussed here (Schearer (1968)). In the following sections we discuss
the mecahnisms for SEOP and the dynamics of the interaction between the polarized alkali, and the polarized noble gas.
41
2.4.1 Optical Pumping of Noble Gas
The spin dependent interaction between the alkali and the noble gas during collision is:
V1 (r ) = γ(r ) N · S + Ab (r ) Ib · S
(2.99)
Here the noble gas spin is denoted as Ib , the alkali spin by S, and the rotational
angular momentum by N. The spin-rotation interaction arises from the magnetic
fields created by the motion of the charges of the colliding atoms. The second term
arises from the hyperfine interaction of the nucleus of the noble gas, and the alkali.
Here the coupling coefficients are strong functions of the inter-atomic separation
r. The subscripts a refer to the alkali atom, while the subscript b to the noble gas
atoms.
The spin-rotation term in the interaction potential leads to alkali spin relaxation. The second term leads to spin exchange between the alkali electron spin
and the noble gas nuclear spin. In the spin temperature regime we can express
the rate equations governing the polarization of the spins by (Walker & Happer
(1997)):
d h Fz i
= − Γ a ( γ ) h Sz i −
dt
Γ a ( Ab )[ǫ( Ib , β b ) hSz i − h Ibz i]
(2.100)
d h Ibz i
= Γb ( Ab )[ǫ( Ib , β b ) hSz i − h Ibz i]
dt
(2.101)
Due to the principle of detailed balance we can say that:
nb Γb ( Ab ) = n a Γ a ( Ab )
where the density of the vapour species is given as ni . For the case of
42
(2.102)
21 Ne,
Ib is
3/2. Thus eq. (2.101) simplifies to:
d h Ibz i
= Γb ( Ab )[3 hSz i − h Ibz i]
dt
(2.103)
Since ǫ is a function of spin temperature this is valid uner conditions of high spin
temperature where Pa ≈ 1. We can readily see that in steady state, if all relaxation
mechanism are suppressed, then the noble gas nuclear spin expectation value is
one third of the alkali spin expectation value. However the nuclear spin of
21 Ne
is 3/2, while the electron spin is 1/2. Taking this into account we can re-write eq.
(2.103) in terms of the polarization of each species respectively as:
3
1
Pb = 3 Pa
2
2
(2.104)
Or, simply Pa = Pb , under conditions of high spin temperature. We see that in
steady state and under the absence of relaxation mechanisms the noble gas polarization equilibrates with the alkali vapour polarization. This argument is valid
for an atom with any nuclear spin value. However in practice the noble gas polarization does not reach the same value as the alkali polarization due to strong
relaxation mechanisms. These include spin destruction collisions, gradient relaxation, and long range magnetic dipolar and quadrupolar field interaction. Typical
polarizations are on the order of 40 − 50% under conditions of high pumping rate.
The equilibrium nuclear polarization can be written as:
Pb = P a
ab
ǫ
Rse
b
ab + 1/T I /Sz
Rse
1 bz
ab =
Where the spin exchange rate Rse
dh Ibz i
dt
rate for neon polarization.
43
(2.105)
= Γb ( Ab ), and 1/T1b is the relaxation
2.4.2 Interaction of polarized alkali with polarized noble gas
The interaction of the polarized alkali and polarized noble gas yields interesting
behaviour. The magnetic field experienced by the alkali is due to both the classical magnetic holding field in which the atoms are present, and the magnetic field
created by the polarized noble gas. The field experienced by the alkali due to the
noble gas is not given by the classical expression of a magnetic dipole field. This
is because collisions between the alkali and noble gas deform the wavefunction of
the alkali and also overlap. The dominant interaction between the alkali electron
spin and the noble gas nuclear spin is described by eq. (2.99).
The hyperfine interaction coefficient arises from the Fermi-contact magnetic
fields of the two atoms (Herman (1965)):
A b (r ) =
8πgs µb
|Ψb (0)|2
3Ib
(2.106)
Due to the deformation of the wavefunction during collision it becomes:
A b (r ) =
8πgs µb
|ηΦ( R)|2
3Ib
(2.107)
The enhancement factor η is the ratio of the wavefunction at the noble gas nucleus
during collision to the unperturbed wavefunction in the absence of noble gas. This
isotropic hyperfine interaction also leads to the frequency shift of the precession
frequencies of both the alkali and the noble gas. The shift is described by the parameter κ0 , where κ0 is the ratio of the magnetic field experienced by the alkali due
to the collision with the noble gas, and that which would arise from a Fermi contact
interaction with no η wavefunction enhancement factor (Schaefer et al. (1989a)).
κ0 =
Z
4πR2 |ηΦ0 ( R)|2 e−V0 ( R)/k B T
44
(2.108)
Where V0 is the spin independent interaction between the Noble gas and the alkali.
Normally this is fit to pseudo-potentials using data from scattering experiments.
The dominant interaction between the alkali nuclear spin and the alkali electron
spin is:
Hhyp = A a Ia · S
(2.109)
The A a term in the isotropic interaction can produce a pressure shift in the hyperfine splitting. This is because the valence electron density is perturbed by the noble
gas resulting in a shift in the energy levels. For the most part optical pumping experiments measure magnetic resonance of Zeeman levels, and are thus not very
sensitive to shifts in the A a parameter. This is an important effect in the operation
of atomic clocks.
The effective magnetic field the alkali experience due to the polarized noble gas
for a spherical volume is:
B=
8πκ0
MP
3
(2.110)
Where M is the magnetization density of a fully polarized noble gas, and P is the
noble nuclei polarization. The alkali atoms precess in the resultant field caused
by both the static magnetic holding field, and the magnetic field produced by the
noble gas nuclei. The motion of the alkali polarization can be described by the
phenomenological Bloch equations (Kornack & Romalis (2002)):
8πκ0
∂Pe
γe
B+
=
Mnoble Pnoble + L × Pe + Ω × Pe
e
∂t
Q( P )
3
ab n
e
+ Rop s pump + + R probe s probe + Rse P − Rrel P /Q( Pe )
∂Pn
= γn
∂t
8πκ0
Malkali Pe
B+
3
(2.111)
ab
× Pn + Ω × Pn + Rse
(Pe − Pn ) − Rrel Pn (2.112)
γe is the gyromagnetic ratio of a free electron, Rrel is the alkali relaxation rate, Pn
ab is the alkali-noble gas spin exchange rate. R
is the nuclear polarization, and Rse
op
45
and R probe are the pump and probe beam pumping rates.
One can write this system of equations in complex form by multiplying the y
component by the imaginary number i, adding it to the x component, and solving
the resulting complex differential equation. For small non-equilibrium excitation
of the spins when there is no transverse magnetic field present we can solve this
set of equations. The linear approximations to the solutions is:
8πκ0
8πκ0
z
z
) P⊥e − Rtot P⊥e + i
)/Q( Pe )
Mnoble Pnoble
Mnoble Pnoble
3
3
(2.113)
′
8πκ
8πκ
0
0
n
z
z
(t) = −iγn ( Bn +
P⊥
) P⊥n − Rtot P⊥n + i
) P⊥n
Malkali Palkali
Malkali Palkali
3
3
(2.114)
′
e
(t) = (−iγe ( Bn +
P⊥
Here the xb, and yb components correspond to the real and imaginary components
of the solution.
One can see that the magnetic fields experienced by the alkali and Noble gas
species differ. This is because in addition to the holding field each species experiences the magnetic field produced by the magnetization of the other sample.
Bze = Bn +
8πκ0
z
Mnoble Pnoble
3
(2.115)
Bzn = Bn +
8πκ0
z
Malkali Palkali
3
(2.116)
The co-magnetometer experiments in this thesis operate near a noble-alkali hybrid
resonance described by the magnetic field compensation point:
n
Bcomp
=−
8πκ0
8πκ0
z
z
−
Malkali Palkali
M Noble PNoble
3
3
(2.117)
Although the alkali is not at zero field, the field is low enough that the alkali remains in the SERF regime. Operation at the compensation point leads to screening
46
of transverse magnetic fields. This will be discussed shortly.
One application of particular interest is the signal dependence of the co-magnetometer
on rotation. Solving the Bloch equations one finds the signal to be (Kornack et al.
(2005b)):
Srot
Pze γe Ωy
=
γn Rtot
γn
γe2 2 Bz
e
1−
Q( P ) − C − 2 Bz + n − D + · · ·
γe
B
Rtot
(2.118)
where
C=
γe Pze Rnse
≈ 10−3
n
γn Pz Rtot
(2.119)
Me Rese
≈ 10−5
Mn Rtot
(2.120)
D=
where Ωy is the angular rotation rate of the apparatus. Pze is the electron polarization, Rnse is the noble gas spin exchange rate, Pzn is the noble gas polarization, Rtot
is the total alkali spin relaxation rate, and Rese is the alkali spin exchange rate. The
signal is defined to be S = R( Pxe ).
For a holding field set to the compensation point:
Pze γe Ωy
S(Ωy ) =
γn Rtot
γn
1−
Q( Pe ) − C − D + O(10−6 )
γe
(2.121)
One can see that the system is sensitive to any rotations about the yb axis. The
experimental implications of this will be discussed later, suffice it to say that this
allows one to operate the coupled system as a gyroscope.
The signals from rotation about the other two axes is much smaller. This is by
a factor of approximately 100 for the xb and 105 for the b
z axes.
The coupled system also exhibits interesting shielding behaviour when set to
the compensation point. The signal becomes insensitive to small applied transverse magnetic fields. When the system is tuned to the compensation point the
noble gas spins are aligned with the holding field. If a small transverse field in47
(a)
21Ne cancels the external field
Bn
(b)
K feels no field
SK
MK
I 21
M 21 Ne
21Ne compensates for
Bx
K feels no change
SK
MK
II 21
Ne
Bn
n
B
M 21 Ne
Ne
B
B
Bn
Bn
Bx
Figure 2.10: When set to the compensation point the polarized noble gas screens
transverse magnetic fields.
teracts with the system the noble gas spins will adjust to align itself with the total
magnetic field. To first order the noble gas cancels out any transverse field. This
suppresses fields transverse to the holding field. Magnetic fields parallel to the
holding field do not tip the alkali since they will not adjust the coupled system.
2.5
Manipulation of polarized noble gas spins, and Magnetic shielding
In experiments where noble gas is polarized it is often necessary to manipulate
the noble gas orientation. Examples of this include performing adiabatic fast passage for determination of alkali noble gas spin exchange cross section (Chann &
Walker (2002)), and nuclear magnetic resonance for determination of κ0 (Stoner &
Walsworth (2002)), and pulmonary imaging (Oros & Shah (2004)). Many optical
pumping experiments benefit from operation in a low magnetic field environment
(Kornack et al. (2008)). This can be achieved by utilizing magnetic shielding. This
suppresses the magnetic field the atomic species experience by several orders of
48
magnitude. In this section we discuss each of these techniques for manipulation of
both the polarized noble gas, and shielding in greater detail.
2.5.1 Adiabatic Fast Passage
In optical pumping experiments it is often useful to flip the noble gas polarization
by 180◦ . The most common technique for achieving this with relatively low loss
of polarization is Adiabatic Fast Passage (AFP). To achieve this one applies a magnetic field perpendicular to the magnetic holding field of the alkali. This field is
then swept in frequency through the magnetic resonance frequency of the alkali
due to the holding field. Alternatively the frequency of the perpendicular applied
field can be held constant, and the strength of the field can be varied. Under appropriate conditions this process can result in the inverting of the alkali spins. To
explain this effect it is necessary to review some dynamics of operators viewed in
a rotating frame.
~ in an initially inertial frame. If one transfers to a frame
Consider the vector A
~ then A
~ transforms as:
which is rotating with frequency Ω
~
dA
dt
!
~
dA
dt
!
=
rotating
=
inertial
~
dA
dt
~
dA
dt
!
!
inertial
rotating
~
−~
Ω×A
(2.122)
~
+~
Ω×A
(2.123)
Now let us consider a spin ~S held with a field B0 in the inertial lab frame. It undergoes Larmor precession according to:
~
dA
dt
!
inertial
49
= γ~S × ~B0
(2.124)
In the rotating frame this becomes:
d~S
dt
!
d~S
dt
!
or,
rotating
rotating
= γ~S × ~B0 − ~
Ω × ~S
(2.125)
!
(2.126)
= γ~S ×
~
~B0 + Ω
γ
The system behaves as if it experiences an effective magnetic field
~Be f f = ~B0 + Ω
γ
(2.127)
in the rotating frame. The spins rotate about this effective field in the rotating
frame.
Let us consider the special case where the applied field ~B1 is rotating with ~
Ω is
equal to the precession frequency ω0 = −γB0 of the alkali. Here we find:
~Be f f = ~B0 + ~B1 + ω b
z
γ
or since ω0 = −γB0 ,
ω
~Be f f = B0 b
z
z + B1 xb + b
γ
~Be f f = B1 xb
(2.128)
(2.129)
(2.130)
When the frequency of the applied transverse field is exactly on resonance with
the precession frequency due to the holding field in the rotating frame, the spins
experience a static transverse field perpendicular to the holding field. This causes
the spins to rotate about the xb and flip.
In the more general case where Ω 6= ω0 one does not completely cancel out
the effect of the holding field as described in the transition from equation 2.129
to eq. (2.130). The atoms still experience a static field B1 xb but also experience a
50
B0
ω
γ
Beff
θ
B1
Figure 2.11: Here one can see that if a transverse field is applied at a frequency
other than the resonant frequency it appears as having both transverse and axial
components in the rotating frame. The relative strength of these components is a
function of the detuning from resonance. The atoms precess around the effective
magnetic field which lies at an angle θ to the holding field. As one sweeps the
frequency of the transverse field B1 from far below resonance to above it the angle
θ goes from 0 to π. Thus we can flip the orientation of the spins in this manner.
field
Ω − ω0
z along
γ b
the holding field axis. See fig.2.11. The alkali now rotate around
the general field ~Be f f in the rotating frame. As the frequency Ω is swept the field
along b
z changes. The angle ~Be f f makes with the holding field is also changed. If Ω
is varied slowly then the spins will follow and continue to precess around the field
Be f f . Using this knowledge we can sweep the field from far below the resonant
frequency to far above the resonant frequency to flip the spins by π radians.
To ensure that the alkali spins follow Be f f one must change the frequency of the
transverse field slowly. Physically this condition is satisfied when the precession
rate is very large compared to the rate at which the direction θ (t) of Be f f is changing (Powles (1958)). This condition is most stringent when the applied frequency
is equal to ω0 or geometrically when θ =
π
2.
So,
dBe f f (t)
θ̇ = 1
B1
dt
51
(2.131)
or,
1 1 dΩ
<< γB1
B1 γ dt
(2.132)
Substituting the condition ω = γB1 which occurs at θ = π/2 we find:
dΩ
<< ω02
dt
(2.133)
to ensure the spins adiabatically follow the Be f f .
One cannot sweep the magnetic field frequency arbitrarily slowly. When one
has a Be f f which is not parallel to the holding field the spins will dephase and lose
polarization after a time T1 . We must sweep the spins faster than a time T1 to retain
spin polarization. Mathematically this corresponds to:
γB1
dΩ
<<
T1
dt
(2.134)
Thus to ensure one can flip the spin by sweeping the frequency of the applied field
one must satisfy:
dΩ
γB1
<< ω02
<<
T1
dt
(2.135)
2.5.2 Nuclear Magnetic Resonance
In order to determine the κ0 magnetic enhancement factor one must determine
noble gas polarization. To do this we utilize the techniques of nuclear magnetic
resonance. As this is quite a broad field only the fundamentals which were applied
in the experiments in this thesis will be reviewed.
Nuclear magnetic resonance (NMR) is the technique by which polarized spins
precessing in a magnetic field are manipulated and detected. These are tipped or
excited so that they possess a transverse polarization. As the transverse magnetization rotates they produces an oscillating magnetic field which is typically detected
52
by inductive pickup coils. To tip the spins one excites them with a magnetic field
pulse which is at the same frequency as the precession frequency of the atoms. If
the excitation tipping field, along the xb direction, is of strength B1 then the field
experienced in the frame rotating at the Larmor frequency of the nuclei is:
~Be f f = ~B0 + ~B1 + ω b
z
γ
or since ω0 = −γB0 ,
ω
~Be f f = B0 b
z
z + B1 xb + b
γ
~Be f f = B1 xb
(2.136)
(2.137)
(2.138)
This argument follows directly from that in the previous section, where we assume
the holding field along the b
z direction. We can see that the nuclei will now precess
along the xb direction. Typically the applied field however is of the form:
~B1 = B1 cos(ωt) xb
(2.139)
which is not equivalent to a field rotating at the precession frequency. However it
can be decomposed into two counter-rotating fields at the precession frequency:
1
~B1 = 1 B1 (cos(ωt) − sin(ωt)) b
σ+ + B1 (cos(ωt) + sin(ωt)) b
σ−
2
2
(2.140)
The σ− component is located far enough off resonance that its effect is negligible in most cases. We will not discuss it but it can be of interest in certain comagnetometers operating in the SERF regime. If the duration of the magnetic field
pulse is for a time t then we see that the angle the spins rotate through an angle
given by:
Θ=
1
γBt
2
53
(2.141)
B0
ω
S
Β1cos (ω t)
θ
Figure 2.12: If an oscillating magnetic field B1 is applied at the atom precession
frequency orthogonal to the holding field B0 then the atoms will become tipped off
axis.
See fig.2.12.
In order to detect the field produced by the precession of the tipped spin one
constructs an inductive pickup coil. To ensure that the signal is larger than the
various instrument noise one utilizes the pickup coil to create a resonant LC circuit.
Here a capicator is placed is parallel with the pickup coil. For an ideal coil of
inductance L, and resistance r the impedance of parallel RLC circuit is given by
(Fukushima & Roeder (1981)):
Ztank =
r − iωL(1 − ω 2 CL − r2 C/L)−1
r 2 + ω 2 L2
(2.142)
The resonant condition occurs when:
1 − ω 2 CL − r2 C/L = 0
(2.143)
However for most systems the resistance of the pickup is quite small. Here we
54
get the resonant frequency to be:
1
ω= √
LC
(2.144)
The impedance at the resonant condition is given by:
Zresonant
r 2 + ω 2 L2
=
r
(2.145)
Typically the resonant frequency of the pickup is tuned by placing a variable capicator in the RLC circuit, and adjusting it.
It is often convenient to define the quantity Q = ωL/r. This is referred to as
the quality factor of the pickup circuit. For an inductive pickup coil the Q factor
is the factor by which the pickup voltage is magnified at the resonance condition
as compared to the case of detecting the precessing spins directly with the pickup
coil, and not using a RLC circuit.
2.5.3 Magnetic Shielding
For sensitive magnetometry experiments it is often advantageous to operate inside magnetic shields. These reduce or eliminate magnetic field contributions due
to the Earth’s field, and laboratory power line 60Hz magnetic noise. This ensures
that the field in which the atoms are held is well known and uniform. These shields
typically are made from ferromagnetic materials, have high permeability, and are
easy to both magnetize and demagnetize. Typical commercial brands are made
from Mu-Metal. Other materials often used in magnetic shield manufacture are
Conetic alloys, and Moly Permalloy. Recently work has been done to observe the
effectiveness of utilizing ceramic ferrite as a shield (Kornack et al. (2007)) which
have lower thermal noise than the Johnson noise of Mu-Metal shields at room tem-
55
perature.
The standard arrangement for magnetic shielding is to make multiple concentric cylinders which are nested within each other. If one performs the calculation
they find that the shielding factor in the transverse and axial direction differ for a
cylindrical shield. The transverse shielding factor is given by (Jackson (1999)):
ST =
µt
Bi
=
B0
2R
(2.146)
where µ is the magnetic permeability of the shield, t is its thickness, and R its
radius. Here B0 is the uniform field outside the shields, and Bi the field inside the
shield. The axial shielding factor is given by (Khriplovich & Lamoreaux (1997)):
SA =
Bi
2µtR1/2
≈
B0
L2/3
(2.147)
where L is the length of the cylinder. See fig.2.13. This approximation is only valid
in the region where 4 < L/R < 80.
Normally one must place holes in magnetic shields for various electronic feedthroughs, power cables, optical beam paths, etc. This decreases the shielding factor
slightly. For a hole of radius r the field perpendicular to the shield surface falls off
as (Khriplovich & Lamoreaux (1997)):
B(l ) = Be−1.5l/r
(2.148)
where l is the distance from the hole.
The shielding from nested cylinders is not simply the product of the individual shielding factors of each layer. This is because the internal shields affect the
boundary conditions used in the previous calculation of the shielding factors. The
56
Magnetic
Shields
Air
Baxial
Figure 2.13: The magnetic fields lines have an affinity for magnetic shields. The
above figure shows how magnetic fields lines from a previously homogeneous
field warp in the presence of magnetic shields. The solution of the Laplace
equation in cylindrical polar coordinates gives a magnetic scalar potential Φ =
(Ck ρ + Dρk )cosθ. The relevant boundary conditions for the magnetic field compok
nents normal and tangential the surface are µk dΦ
dρ |rk = µk+1
1 dΦk+1
rk dθ |rk
dΦk+1
1 dΦk
dρ |rk , and rk dθ |rk
=
The magnetic field pattern can be constructed by solution of the Laplace
equation, subject to application of the boundary conditions described above.
57
transverse shielding factor for nested cylinders is given by (Sumner et al. (1987)):
ST = SnT
n −1
∏ SiT
i
"
1−
Ri
R i =1
2 #
where the script i refers to each individual layer. The 1 −
(2.149)
Ri
R i =1
2
compression
of internal flux (volume loss) in the region between layers. The axial shielding is
given by:
SA =
SnA
n −1
∏
SiA
i
L
1− i
L i +1
(2.150)
Here the 1 − LLi term reflects the reduction in volume or compression of flux along
i +1
the cylinder length.
Before use the magnetic shields must be de-gaussed. This necessitates reduction of the magnetization of the shields to zero. To accomplish this one passes a
high alternating current, normally 50 amp-turns, through the shields to saturate
them. The current in the shields is then slowly reduced to zero. The idea is that
when one passes a high current through the shields the saturation is symmetric on
current reversal. If one then reduces the current slowly, much slower than the alternating current frequency, one can eliminate the magnetization when the current
becomes zero.
58
Chapter 3
Nuclear Spin Gyroscope
Sensitive gyroscopes are utilized in applications ranging from inertial navigation,
and studies of the Earth’s rotation, to tests of general relativity (Stedman (1997)). A
variety of physical principles have been employed for rotation sensing. These include mechanical gyroscopes, the Sagnac effect for photons (Stedman (1997))(Andronova & Malykin (Andronova & Malykin)) and atoms (Gustavson et al. (1997))(YverLeduc (Yver-Leduc)), the Josephson effect in superfluid 4 He, and 3 He (Avenel et al.
(2004)), and nuclear spin precession (Woodman, Franks & Richards (Woodman
et al.)). While mechanical gyroscopes operating in low gravity environments remain so far unchallenged (Buchman et al. (2000)) there is much competition in the
field of compact gyroscopes operating in Earth’s field.
We have developed a nuclear spin gyroscope (Kornack et al. (2005a)) based
on the co-magnetometer arrangement described earlier in this work (Allred et al.
(2002)). As mentioned earlier the signal from the atom co-magnetometer has a
dependence on the rotation of the apparatus, thus it can be utilized as a gyroscope.
Though the entire dynamics of the coupled alkali-noble gas system is complicated
it is useful to describe the system physically before delving into mathematics. The
work in this chapter can also be found in Kornack et al. (2005a).
59
3.1
Co-magnetometer Gyroscope Implementation and
behaviour
In the co-magnetometer arrangement the K alkali spins are polarized with a pump
laser. Through spin exchange collisions the nuclear spins of the 3 He noble gas are
polarized parallel to this. Let us imagine the cell which contains noble gas to be in
an inertial reference frame. If we rotate the apparatus the alkali spins will quickly
be re-pumped along the new pump laser orientation since the T1 of the alkali is on
the order of 30ms. The noble gas then precess around the net magnetic field and
becomes aligned with the holding field in a time T2 of ≈ 100s. We will focus on
the interesting dynamics on a short time scale.
Initially if the co-magnetometer is properly zero-ed, and tuned to the compensation point then in the steady state arrangement the noble gas nuclei experience
a net magnetic field parallel to the nuclear spin due to the magnetic compensation
field. When the apparatus is rotated the compensation field is now at an angle to
the nuclear spins causing them to precess about it. As the nuclear spins precess
the orientation of the net magnetic field which the alkali experience now changes.
This causes the potassium to rotate about the new orientation of the net magnetic
field. Consider the component of the magnetic field perpendicular to the pump
and probe beam optical axis which is produced by the noble gas. This field causes
the alkali to precess in the plane of the pump and probe beam. It is this alkali precession that we monitor with an off resonantly tuned probe beam. See eq. (2.121).
Using Green’s functions for the linearized Bloch equations one can show that
the integral of the co-magnetometer signal is proportional to the total rotation angle about the yb axis. This is independent of the time behaviour of Ωy . The angular
60
Position sensors
P u m p B ea m
H i gh P ower
D i ode L aser
?/4
P i ezoel ect r i c S t ack
I mmobi l e B l ock
Floating Optical Table
Lock-in
Amplifier
Photodiode
Analyzing Polarizer
Magnetic Shields
Field Coils
Hot Air
Cell
SK
MK
Probe Beam
Single Freq.
Diode Laser
M 3He
Polarizer
I 3He
Bz
Faraday Pockel Cell
Modulator
x
y
z
Figure 3.1: A schematic of the co-magnetometer being implemented as a gyroscope. Note the non-contact position sensors used to detect the rotation, and the
piezo stack used to force the apparatus to oscillate
frequency of the alkali can be related to the co-magnetometer signal by:
Ωy = γg S
(3.1)
where S is the signal from the co-magnetometer in units of magnetic field and γg
is given by:
γg ≈
1
Q( Pe )
−
γn
γe
−1
(3.2)
The apparatus was driven to rotate by a piezo stack placed between the optical
table upon which the co-magnetometer sits and a heavy immovable concrete block.
This is depicted in fig.3.1, and fig.3.2.
Six non-contact displacement sensors were used to monitor the table orientation. They were mounted to the posts at the base of the optical table. They operate by measuring the capacitance between the sensor and a target metallic plate.
The target plates were mounted to the floating portion of the optical table. The
sensors was calibrated and found to give a linear response over displacement am61
Cell
Spin polarization
Pump Beam
Probe Beam
Photodiode
Table rotation
Piezo Stack
y
Position
Sensors
z
Immobile
Block
y
x
x
z
Figure 3.2: Alternate side view of the gyroscope configuration for the comagnetometer
5.0
Signal (volts)
4.5
4.0
3.5
3.0
2.5
2.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Horizontal displacement from target (cm)
Figure 3.3: In-situ calibration of the non-contact displacement sensors for determination of absolute rotation.
plitudes of a few centimeters. For comparison during operation the amplitude of
displacement was on the order of millimeters. The sensors were each individually
calibrated in-situ. See fig.3.3.
In order to convert the co-magnetometer signal to a gyroscope we must integrate the co-magnetometer signal. Using eq. (3.1) we convert to angular units.
The resulting fit of the co-magnetometer gyroscope signal to the orientation signal
from the non-contact displacement sensors in shown in figure 3.4
The gyroscope signal agrees with the position measurement to within the 3%
calibration accuracy. The gyroscopic signal has also been shown to be insensitive
62
50
0
0
− 50
− 20
E ff ecti v e F i el d ( f T )
R otati on ( µ r ad / sec)
100
20
− 100
0
2.5
5
7.5
10
12.5
Time (s)
0.02
25
0.015
20
15
0.01
10
0.005
F i el d ( fT / H z1/ 2 )
A ngl e R andom W al k ( degr ees/ hour 1/ 2 )
Figure 3.4: Comparison of co-magnetometer gyroscope signal to displacement sensor signal with no free parameters. The solid line depicts the co-magnetometer
signal, and the dashed line the signal from the position sensors.
5
0
0
200
400
600
800
0
1000
Frequency (hour − 1 )
Figure 3.5: Fourier transform of the noise spectrum of the comagnetometer gyroscope. The discrete noise peaks are an artifact caused by the periodic zeroing
routines for the co-magnetometer
to the other two components of the angular velocity, and only depends on the
angular velocity vector perpendicular to the plane containing the pump and probe
beams.
Using the relation between the magnetic field measurement of the co-magnetometer
and the rotation measurement of the gyroscope one can calculate the noise spectrum of the gyroscope from previous magnetic noise measurements of the comagnetometer. As one can see from fig 3.5 the noise spectrum of the gyroscope
√
√
is nearly flat at 1.0ft/ Hz, or translated to angular units 1.4 × 10−5 rad/ hour for
frequencies above 400 hour−1 . At frequencies below this one sees a clear 1/ f noise
63
F i el d S u p p r essi on F actor
100
dBy/dx
dBy/dy
10− 1
dBy/dz
dBx/dz
−2
10
dBz/dz
10− 3
10− 4
0.1
0.2
0.5
1
2
5
10
Frequency (Hz)
Figure 3.6: Suppression of an applied magnetic field gradient by the comagnetometer compared to that of a non-compensating magnetometer. Colored
points refer to measurements made with square wave modulation instead of sinusoidal modulation.
dependence with a 1/ f noise knee at 0.05Hz. The noise of the gyroscope in magnetic units is much less than the magnetic Johnson contribution from the magnetic
shielding. This is because the co-magnetometer acts as a magnetic field suppressor.
This is discussed in more detail in section 2.6. The magnetic field is suppressed
even though the alkali metal and Noble gas have different spatial distributions.
(see fig 3.6). The reason for this behaviour is because the noble gas diffusion rate
is much lower than the nuclear spin precession rate. That is γn Bn >> R D >> Rnsd ,
where Rnsd is the nuclear spin destruction rate. The magnitude of the polarization
of the noble gas is constant in value, and orients itself parallel to the local magnetic
field. Thus the noble gas is able to cancel a non-uniform external field on a point
by point basis.
An oscillating magnetic field was applied and effectively shielded by the comagnetometer. This is shown in fig 3.7. In addition, using the linearized Bloch
equations one can show that the rotation angle created by a magnetic field transient
is zero as long as the spin polarizations are not rotated by a large amount during
the transient. Fig 3.8 shows the response of the gyroscope to a magnetic field transient. The co-magnetometer demonstrates a reduction of the spin rotation angle
by a factor of 400 as compared to that of a standard potassium co-magnetometer.
64
Bx
By
10
Bz
0.1
1
0.01
M easu r ed F i el d ( p T )
F i el d S u p p r essi on F actor
1
0.1
0.001
0.1
0.2
0.5
1
2
5
10
Frequency (Hz)
60
0.3
40
0.2
20
0.1
0
A ngl e ( r ad )
M agneti c F i el d ( p T )
Figure 3.7: The co-magnetometer suppresses magnetic fields. Thus the contribution from the Johnson noise of the magnetic shields is greatly reduced. The field
suppression in the xb,b
y, and b
z directions are measured relative to that of a K magnetometer. The solid line represents the theoretical prediction.
0
− 20
− 0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
Figure 3.8: Response of co-magnetometer(dashed red line) to a magnetic field transient(solid line). The total rotation angle(blue dashed line) is proportional to the
integral of the co-magnetometer signal. It is also much smaller than the rotation
angle from a K magnetometer(dashed black line).
65
3.2
Effect of Experimental Imperfections on Gyroscope
Performance
Gyroscope performance can be affected by a number of experimental imperfections of the system. One such imperfection is that arising from not properly zeroing the magnetic field on the co-magnetometer. The only component of the magnetic field or light shift which contributes to the signal in first order is the Bx field.
It causes a false signal of (Kornack et al. (2005a)):
n
e
)/Bn
S( Bx ) = Bx Pze (Cse
+ Cse
(3.3)
e
Cse
= ( Rese Pzn )/( Rtot Pze )
(3.4)
where
is the electron spin exchange correction, and
n
Cse
= (γe Pze Rnse )/(γn Pzn Rtot )
(3.5)
is the nuclear spin exchange correction factor which arise from solving eq. (2.111).
e ≈ 10−2
Substituting the measured values for Pze ,Pzn , Rese , Rnse , and Rtot we find Cse
n ≈ 10−3 . Since both C e and C n are small this field dependence is heavily
and Cse
se
se
suppressed by a factor of ≈ 105 . Pze is the alkali polarization,Pzn the noble gas
polarization, Rese is the alkali metal-noble gas spin exchange rate for an alkali atom
and, Rnse is the same for a noble gas atom. Rtot = R pump + Rese + Resd + R probe
Misalignment of the pump and probe beam so that they are not orthogonal also
causes a systematic rotation signal. If they are misaligned an angle α away from
90◦ we measure a signal:
S = αR pump /Rtot
66
(3.6)
Under typical operating conditions of the gyroscope a misalignment of 1µrad gives
a false signal of 10−8 rad/s. However this can be corrected for because a true rotation signal has no dependence on the pumping rate. One could correct for this by
varying the intensity of the pump beam, and aligning the probe beam until there is
no longer a gyroscope signal at the frequency at which the pump intensity is being
varied.
One can also measure a false signal if the probe beam polarization is not fully
linearly polarized, but has a small circular polarization component. This gives a
signal of:
S = sm Rm /Rtot
(3.7)
where the circular polarization of the probe beam is given by sm , and Rm gives
the pumping rate of the probe beam. This pumping by the probe beam can be
eliminated by zeroing the probe beam light shift. This will be described further in
the next section.
Other experimental imperfections only contribute to the gyroscope signal in
second order. For comparison the signal from an improperly zeroed field By is
given by:
S=
γe By Pze
e
n
( Bz − ( Bz + Lz )Cse
− (2Bz + Lz )Cse
)
Bn Rtot
(3.8)
where Bz is the amount by which the b
z field is tuned away from the compensation
point. The Effects of imperfections in a Rb-Ne co-magnetometer are discussed in
Sec.6.4.
3.3
Zeroing the Co-magnetometer Gyroscope
In order for the co-magnetometer signal to be properly calibrated it must be zeroed. That is the net magnetic field along both the xb, and yb directions must be
67
zero. The light shifts must be removed, and the b
z field must be tuned to the
compensation point. To do this we employ a quasi-static modulation technique.
That is we modulate certain components of the field at very low frequency. That
is a frequency much lower than the resonance of the noble gas. In the 3 He comagnetometer this corresponds to a frequency of roughly 7 Hz during normal operation conditions. This frequency is determined by the field experienced by each
atom species when the co-magnetometer is brought to the compensation point.
The steady state solution to the coupled Bloch equation which govern the comagnetometer gives the co-magnetometer signal in the steady state regime in units
of magnetic field as:
Ωy sm Rm + αR p
+ Bz
+
S = Ly +
γn
γe Pze
By
γe
− Lx
Bn
Rtot
γe
+
Rtot
Bx Bz ( Bz + Lz )
− L x Lz
Bn
(3.9)
We shall consider the zero-ing of each component of the magnetic field or light
shift separately and in greater detail.
If one were to calculate the dependence of 3.9 on By they would find:
δS
∝ Bz
δBy
(3.10)
We see that the dependence of the signal is directly proportional to the Bz component of the field. So if we modulate the By component of the field we observe
a modulation of the signal at the same frequency. Thus we can zero the Bz component by modulating the By field, monitoring the signal, and varying Bz until
the modulation in the signal disappears. This corresponds to the point where the
compensation field cancels out the field contribution from the magnetization of the
atoms, and any residual field due to the magnetic shielding.
Once the field has been tuned to this compensation point one can use a similar
68
method to zero the By field. One sees that the signal dependence on Bz is:
By
γe
δS
∝ n + Lx
δBz
B
Rtot
(3.11)
We ignore the contribution of the Bx Bz term here as it is the product of two small
numbers and is suppressed compared to the other terms in the above expression.
We see that the modulation of Bz and readjusting the By field will now zero a linear
combination of the By field and the lightshift Lx . We independently zero Lx as
well. This will be discussed later. Thus to zero By we iterate the modulation of Bz
followed by independent zeroing of Lx .
To zero Bx we take a slightly different approach. One can write the dependence
of the signal on Bz as:
δ2 S
∝ Bx
δBz2
(3.12)
Thus we can eliminate Bx if we modulate the second derivative of Bz . To do this
we asymmetrically modulate Bz between a zero, and non-zero value. If one were
to perform this modulation symmetrically one would simply be repeating the earlier procedure used to zero By . Although this response would be different since
the Bx Bz term would now be of significant size compared to By since the latter
component has been zeroed. This function would also be dependent on Lx and By .
Once the magnetic field components have been zero-ed we zero the contribution of lightshifts. We consider the Lx Lz term of eq. (3.9). We see:
δS
∝ Lx
δLz
(3.13)
δS
∝ Lz
δL x
(3.14)
Thus we can modulate the appropriate component of the light shift to zero the
other. The light shift of a laser goes as the frequency of detuning, modified by the
69
Voight profile, and the degree of circular polarization. To modulate the light shift of
the pump beam Lz one modulates the frequency of the laser. This is because tuning
the pump to be directly on resonance maximizes pumping efficiency. To modulate
the lightshift of the probe beam Lx one varies the degree of circular polarization.
To do this we attempt to cancel any birefringence in the beam before it strikes the
cell. There are a few methods to do this including placing a Pockel cell in the beam
path or a plate of stressed glass. The stress of the glass can be varied by using a
piezo to compress it. This compression can be modulated.
One must also ensure that the pump and probe beam must be properly orthogonalized. The co-magnetometer signal does depend to first order on the product
of the angle misalignment α and the pumping rate R pump . However in practice
one cannot make the pumping rate high enough to make it the only term which
contributes to the signal. The signal dependence on the pumping rate goes as:
R pump
Ωy R pump
δS
∝ αK
+ γe
δR pump
Rtot
γn R2tot
(3.15)
where K is a factor for calibrating the misalignment angle to signal units. To vary
the pumping rate we feed the pump beam on maximum intensity through two
crossed polarizers. In between these polarizers we place a liquid crystal waveplate.
By varying the retardance of the waveplate we are able to change the polarization
of the beam between the polarizers, and adjust the intensity of the pump upon
exiting the polarizers. Thus we can vary the angle α until the dependence on the
signal vanishes.
3.4
Co-magnetometer Gyroscope Sensitivity
The fundamental sensitivity of the gyroscope is accurately described by the spin
projection noise of the co-magnetometer. The measurement uncertainty of the co70
magnetometer is dominated by the noise from the alkali metal spins. In terms of
the angular frequency this can be expressed as:
γn
δΩy =
γe
r
Q( Pe ) Rtot
nV
(3.16)
where n is the alkali metal density, and V is the measurement volume.
Currently we operate the gyroscope with a K-3 He co-magnetometer. The fun√
damental sensitivity for this is δΩy ≈ 1.2 × 10−8 rad/s/ Hz. This is roughly 50
√
times lower than the realized sensitivity of δΩy ≈ 5.0 × 10−7 rad/s/ Hz. This discrepancy can be attributed to the large amounts of angular noise contributed from
the probe beam laser.
By looking at eq. (3.16) we see that the fundamental sensitivity is a function of
the ratio of the noble gas and alkali gyromagnetic ratios. By switching to a K-21 Ne
co-magnetometer should instigate an immediate improvement in the sensitivity
by a factor of roughly 10, since
21 Ne
has a gyromagnetic ratio roughly an order
of magnitude smaller than that of 3 He. For a cell with 10cm3 volume, K density
of 1014 cm−3 , and
density of 6 × 1019 cm−3 this would imply a fundamental
√
sensitivity of δΩy ≈ 2.0 × 10−10 rad/s/ Hz.
21 Ne
To be useful for applications such as navigation, a gyroscope must be small
and portable. The most widely used gyroscope for navigational purposes is the
fiber-optic gyroscope. After nearly two decades of improvement the sensitivity of
these gyroscopes have approached their fundamental sensitivity. Compact state
√
of the art fiber-optic gyroscopes have sensitivity of δΩy ≈ 2.0 × 10−8 rad/s/ Hz
(Sanders et al. (2000)). New compact interferometer gyroscopes using cold atoms
√
with a shot noise sensitivity of δΩy ≈ 1.4 × 10−7 rad/s/ Hz (Canuel et al. (2006))and
√
δΩy ≈ 5.0 × 10−9 rad/s/ Hz (Müller et al. (2007)). Another newly proposed
gyroscope is one which operates with MEMS technology. These hold promise if
71
0.6
A ngl e ( d egr ees)
0.4
0.2
0
− 0.2
− 0.4
Constant drift of 0.1 deg/h
− 0.6
0
1
2
3
4
5
Time (hours)
Figure 3.9: Long term drift of gyroscope
they continue to be developed in the future. Current designs have recently overcome the extreme sensitivity to temperature drift past MEMS groscopes exhibited.(Trusov et al. (2008)). However the current sensitivity of temperature insen√
sitive MEMS gyroscopes are δΩy ≈ 5.0 × 10−4 rad/s/ Hz (Trusov et al. (2008)).
This renders them unusable for navigational applications in the near future.
The compact co-magnetometer gyroscope is competitive with the methods mentioned above. The measurement volume is roughly 10 cm3 , though the present
implementation occupies a square approximately 2m to a side. This setup can
be miniaturized. In fact many of the techniques found in the miniaturization
of atomic clocks can be used to minituarize the gyroscope (Knappe et al. (2006),
and Knappe (2004)). The lasers can be made more compact quite easily. In fact
miniaturizing the magnetic shields improves their performance. The dominant
source of long term drift in the co-magnetometer is due to temperature drift in the
system. Fig3.9 shows the long term drift of the gyroscope. This should also improve dramatically upon minutuarization. The co-magnetometer gyroscope sensitivity is competitive with larger commercial gyroscopes. For comparison gyroscopes based on the Sagnac effect (Stedman (1997)) have achieved sensitivities of
√
δΩy ≈ 2.0 × 10−10 rad/s/ Hz using a ring laser with an enclosed area of 1m2 , and
72
√
δΩy ≈ 6.0 × 10−10 rad/s/ Hz using an atom interferometer with path length of
2m (Gustavson et al. (2000)). One gyroscope which has a much greater sensitivity
is Gravity Probe B. However the wonderfully low drift which it demonstrates, significantly decreases when operated under Earth’s gravity. This makes it much less
suitable for terrestrial based application.
73
Type
74
Large Scale (∼2 m)
Ring Laser Gyro (CII)
Atom Interferometer (Yale)
Intermediate Scale (∼50 cm)
Mechanical (Gravity Probe B)
Superfluid 3 He (Orsay)
Atomic Interferometer (HYPER)
Atomic Fountain (Paris)
Atomic Spin ‘NMRG’ (Litton)
Small Scale (∼10 cm)
Fiber-optic Gyro (Honeywell)
Atomic Spin (This work)
Miniature Scale (< 1 cm)
MEMS (CMU)
Realized
Sensitivity
√
rad/s/ Hz
Projected
Sensitivity
√
rad/s/ Hz
Drift
Citation
rad/hour
2.2 × 10−10
6.0 × 10−10
—
2.0 × 10−10
—
1.3 × 10−4
Stedman (1997)
Gustavson et al. (2000)
—
1.4 × 10−7
—
—
2.9 × 10−6
—
3.0 × 10−10
2.0 × 10−9
3.0 × 10−8
—
3.0 × 10−14
2.1 × 10−5
—
—
9.0 × 10−4
Buchman et al. (2000)
Avenel et al. (2004)
Jentsch et al. (2004)
Yver-Leduc (Yver-Leduc)
Woodman, Franks & Richards (Woodman et al.)
2.3 × 10−8
5.0 × 10−7
—
2.0 × 10−10
1.7 × 10−6
7.0 × 10−4
Sanders et al. (2000)
3.5 × 10−4
1.8 × 10−4
0.5
Xie & Fedder (2003)
Table 3.1: A survey of gyroscope performance.
Chapter 4
Initial tests of an alkali-Neon
co-magnetometer
The main objective of these experiments on
21 Ne
is to ultimately create a neon
co-magnetometer which can be utilized for experiments on tests of fundamental
symmetries, and for deployment as a sensitive gyroscope. We describe construction of a 21 Ne co-magnetometer and investigate the practical problems limiting its
performance.
4.1
Magnetometer setup
The co-magnetometer operates with orthogonal pump and probe beams. These
are both distributed feedback lasers (DFB). These lasers have a reflection grating
etched on the diodes themselves providing a much more stable single frequency
emission. The pump beam has a power of 20 − 40mW, passes through the typical
λ/4 plate and is expanded to a cross section of 3cm2 . It is tuned to the D1 resonance
of K. The probe beam is linearly polarized and passes through the cell and is split
by a polarizing beam splitter into two photodiode detectors. It has a power of
75
Figure 4.1: Experimental setup of Ne Magnetometer
10mW and is tuned 0.2nm away from the D1 resonance of K. The probe beam
cross section is 1.25 × 1.25cm2 defined by a mask. See fig. 4.1.
The cell is filled with a K, 1.6 atm 21 Ne, and 60torr nitrogen mixture in a Pyrex
cell of volume 8.0cm3 . It is heated to 180C◦ . The cell is encased in a glass oven.
The cell is heated via a hot air line which feeds into the glass oven.
The glass housing is placed inside a set of 4 concentric Mu-metal magnetic
shields, which sits upon a standard optical table. Multiple turns of wire run inside the magnetic shields for de-gaussing purposes. These are only connected and
run when the magnetometer in not under operation. Inside the magnetic shields
are sets of Helmholtz coils, and cosine windings wound around a G−7 frame for
magnetic field generation. The internal current used to drive these internal magnetic fields is created by a custom current source. It is based on a mercury battery
voltage reference with a FET input stage followed by an op-amp and transistor
output stage (Baracchino et al. (1997)).
76
N eon P ol ar i zat i on ( ar b. )
0
100
200
300
400
500
600
Time (s)
Figure 4.2: T2 time of ≈ 14minutes for Neon polarization when operating away
from the compensation point in the co-magnetometer configuration.
4.2
Neon Polarization Measurements and Preliminary
Neon Co-Magnetometer data
Neon nuclei were polarized by optically pumping K vapour. The compensation
point for neon was approximately 250µG, which corresponds to a neon magnetization of 8µG, or 0.08% polarization. We have also tipped the neon spins with a
uniform tipping field which was fed into cosine windings. This yielded a T2 time
of approximately 14minutes at low field. See fig.4.2. We have also been able to
show that the K-Ne co-magnetometer shares the same transient response, that is
to compensate for external magnetic transient fields, as the K-He comagnetometer.
See fig.4.2. In short we have demonstrated operation of a K-Ne co-magnetometer,
albeit one with low Ne polarization.
4.3
Influence of Quadrupole collisions in Polarizing
neon nuclei
We were also able to measure the T1 time of the spin exchange optically pumped
neon, as both a function of cell pressure and temperature. It seems that the dom-
77
1
500
0
0
-1
-500
-2
0
50
100
150
200
T ransverse M agneti c F i el d ( pT )
1000
21
N e-K gy roscope si gnal ( V )
2
-1000
Time (sec)
Figure 4.3: Compensation behaviour of the K-Ne comagnetometer to an externally
applied magnetic transient field.
inate limiting factor in achieving large neon polarization is due to quadrupolar
relaxation. These occur during neon-neon collisions when large electric field gradients are created and couple to the quadrupole moments of the respective nuclei
causing depolarization. We believe this is the main cause of nuclear relaxation due
to the following reasons.
First, the T1 time of the longitudinal neon polarization is of the form:
T1 =
1
Rse + Rrelax
(4.1)
So, by knowing the Rse and Rrelax one should be able to reproduce the T1 . Where,
Ne− Ne
Rrelax = R gradient + σsd
[ Ne]ν
and
R gradient = D Ne
where
2
D Ne = 0.79cm /s
78
√
∇B
B
2
1 + T/273.15
Pneon /1amagat
(4.2)
(4.3)
(4.4)
P o l a r i z a t i o n
( a r b .
u n i t s )
12
10
8
6
4
2
0
0
100
200
300
400
500
Time (minutes)
Figure 4.4: T1 of 105 minutes for a 1.6atm cell of Ne at 170C◦ .
For a 6.5atm cell of K-Ne we have a T1 time of 34minutes. Using the theoritical
value of the spin exchange rate and estimating the gradients to be 50µG/cm we
expect a T1 of approximately 140minutes. The gradient calculation was estimated
by modeling the cell as a uniformly polarized cube of magnetization 320µG.
Combining Rse , Rsd and the contribution due to magnetic gradients we still cannot reproduce the observed T1 times. We attribute the discrepancy to a quadrupolar relaxation mechanism. Furthermore we varied the spin exchange rate by measuring T1 at various temperatures without observing a significant change in the T1
time. In fact by increasing the temperature of the 1.6atm cell from 170C◦ to 190C◦
we only observe the T1 change from 105 to 97 minutes. The K density varies by
a factor of over 2 in this temperature range, which would have had a more dramatic effect on the T1 if this were the leading contribution to T1 . It was difficult to
raise the temperature above 190◦ because the optically thick cell was not uniformly
polarized by the pump beam and caused a lower neon polarization. We are further supported by the fact that cells with higher neon pressures have significantly
lower T1 times. For instance at 170C◦ the 6.5atm cell has a T1 of 35.1 ± 1.9minutes
79
whereas the 1.6atm cell is 104.7 ± 0.7minutes. See fig.4.4.
The contribution from neon quadrupolar relaxation can be quantified by measuring the T1 times of cells with different pressure. See fig 5.7. The slope of this fit
is 214 ± 10min.atm. This is discussed in more detail in sec.5.4. Grover (1983) has
also polarized neon and suggests that neon quadrupolar relaxation is the dominant
source of neon relaxation. His data suggests a relaxation of 240min.atm which is
consistent with our data. Although he performed his experiments with Rb instead
of K most relevant exchange cross sections do not vary dramatically between the
two alkali species. We can still attribute the dominant relaxation to be dependent
on the neon properties rather than that of the alkali.
Finally we investigated the effect of filling the cells with both
21 Ne
and 4 He.
Originally this was done to broaden the optical resonance of the K line and decrease the optical thickness of the cell to ensure uniform polarization of the cell.
This was to ensure that the low T1 of the cell was not due to relaxation and inefficient pumping caused by a non-uniformly polarized cell. We filled the cell with
1.73atm of neon and 3.24atm of helium. We measured a T1 of 40.7 ± 1.4 minutes.
This leads us to believe that the effect of Ne-Ne collisions may be comparable to
that of Ne-He collisions.
4.3.1 T1 measurement of Neon
The inference that quadrupolar relaxation is the dominant relaxation mechanism
for neon is dependent on the accurate measurement of the T1 time. To put this on
a firm footing it would be prudent to discuss the measurment of T1 in more detail.
Determining the spin dynamics of the co-magnetometer and determining the T1
data is not trivial. A problem arises because the T2 time is comparable to the T1
time for the measurements made inside magnetic shields.
The large T2 makes measuring the T1 time difficult. To measure the T1 we first
80
tipped the neon spins by a small angle using field coils wound in a cosine winding arrangement within the magnetic shields. As a result of tipping the spins the
Faraday rotation measured by the probe beam became modulated at the precessional frequency. The signal from the photodiode subtracting amplifier was fed
into a Labview program which fit the data and calculated the precessional frequency, amplitude, and decay constant. These can be utilized to calculate the T2
time, and polarization. However when the T2 time is large and comparable to T1
the cell would significantly increase in polarization during the decay of the transverse polarization. This would not make it possible to accurately determine the T1
time. Instead we quenched the transverse oscillations to eliminate this potential
problem.
To quench the oscillations we first partially polarize the sample. Subsequently
the Faraday rotation signal from the detection system was inverted and used to
generate an oscillating magnetic field in a direction orthogonal to both the probe
and pump directions. This caused a nonlinear response which flipped the spins
to make them anti-aligned with the pumping direction. Subsequently a tipping
pulse was applied, and data was recorded for 60 seconds. Then the signal from the
detection system was again fed into the field coils. This then served to quench the
transverse oscillations.
Let us describe this process in more detail theoretically:
~S = a sin(φ(t)) [ xb cos(ωt) − yb sin(ωt)] + a cos(φ(t))b
z
(4.5)
where a is equal to the magnetization. The quenching field can be expressed as:
i
h
~Bquench = κ ~S · xb yb
(4.6)
Here κ is a parameter with units of s−1 which is dependent on factors such as the
81
gain of the coils, and magnetic moments of the spins. So the total magnetic field
acting on the spins in the laboratory frame is:
i
h
~
~B = ~B0 b
z + κ S · xb yb
(4.7)
~B = ~B0 b
z + κa sin(φ(t)) cos(ωt)yb
(4.8)
~Brotating = κa sin(φ(t)) cos(ωt)yb
(4.9)
If we transfer to the frame rotating at the neon precessional frequency this transforms to:
Similarly the behaviour of our spins can be described as:
d~S
= γ~S × ~Binertial
dt inertial
(4.10)
d~S
= γ~S × ~Brotating
dt rotating
(4.11)
or,
d~S
= γ~S × κa sin(φ(t)) cos(ωt)yb
dt rotating
(4.12)
We are interested in the behaviour of the b
z component of the spin. This becomes:
d~Sz
dφ
= − a sin(φ(t))
= γκa2 sin2 (φ(t)) cos2 (ωt)yb
dt
dt
(4.13)
Solving this we find the behaviour of the spins as a function of time:
φ(t) = 2 tan
−1
tan(φ0 /2)exp −κ
t
sin(2ωt)
+
2
4ω
(4.14)
where φ0 is the initial tip angle. One can see that in the case of positive κ the
quenching field aligns the spins with the magnetic holding field during equilib-
82
Angle (degrees)
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
Time (s)
Angle (degrees)
Figure 4.5: Theoritical simulation of the noble gas spin. The quenching field eliminates transverse oscillations and aligns the spins with the holding field for positive
κ.
175
150
125
100
75
50
25
0
0 2 4 6 8 10 12 14
Time (s)
Figure 4.6: Theoritical simulation of the noble gas spin. The quenching field flips
the spins so that the are anti-aligned for negative κ.
rium. see fig 4.5. For negative values of κ we see that the spins align anti-parallel
to the magnetic holding field. That is the equilibrium value of the azimuthal angle
is π radians, regardless of the initial tip angle φ0 . See fig 4.6
As time goes on and the spins are re-pumped along their original direction
the polarity of the quenching pulses must be flipped so as not to re-flip the spins
to their anti-aligned state again. A Labview program was utilized to control this
entire data taking process and monitor the spin orientation relative to the magnetic
holding field.
83
4.4
Improving Magnetometer Sensitivity
A number of technical schemes for improving the magnetometer noise have been
investigated before the final alkali-neon comagnetometer prototype is built. The
first we discuss is eliminating spurious signal from the gyroscopic signal.
4.4.1 Removing Birefringence and false Faraday Rotation signals
Birefringence in the probe beam can cause a false signal. If there is any degree of
circular polarization in the probe beam the system will additionally pump along
the probe beam axis and introduce light shifts into the system. To reduce this
effect we have utilized a cubic cell shape rather than the typical spherical cell. For
a spherical cell there is no dichroic effect produced for a diametrically traversing
beam. For a spherical cell the light experiences some dichroism as varying amount
of light are reflected off the cell wall for light polarized in the plane of reflection,
and perpendicular to the plane of reflection. Random walk of the beam off centre,
possibly due to convection currents of air, would induce some degree of circular
polarization. This is not an issue for a cubic cell since the surfaces are planar.
In a spherical cell different segments of the probe beam travel varying distances through the cell. Therefore segments of the probe beam experience different
amount of Faraday rotation. This effect is also eliminated for a cubic cell to first
order.
Circular birefringence can also be induced from reflections with mirrors, and
via transmission through various optical elements. This can particularly be the
case for stressed lenses, and mirrors. To eliminate this problem the probe beam
has been forced to travel through a variable waveplate. The variable waveplate
consists of a nematic liquid crystal with polar molecules. As an electric field is
applied the polar crystals orient themselves along the electric field axis. The popu-
84
lation which is oriented along the field is a function of the electric field magnitude.
When the probe beam enters the liquid it will experience different indices of refraction along the direction of the electric field and perpendicular to it, inducing
a tunable phase lag. A square wave function generator to control this retardance
and to eliminate or cancel the circular polarization of the beam has been built.
It seems though that utilizing a stressed plate of glass to alter the birefringence
gives a better noise performance, and will most likely be used in the future neon
co-magnetometer.
4.4.2 Controlling and Monitoring the Laser stability
We were unable to obtain DFB lasers which operated at the Potassium D1 frequency. Thus we had to cool commercially available Eagle Yard laser diodes to
−30◦ C in order to tune it to the required frequency. To achieve this we attached
two thermoelectric coolers (TEC’s) in series to the laser diode baseplate. These
were in turn connected to a large heat sink which was cooled with a continuously
run chilled water supply cold plate. Originally the setup was cooled via a mechanical fan, but the level of measurement noise induced by vibration of the air against
the heat sink was unacceptable.
There is the additional complication that the dew point in the lab is approximately 12◦ C. Thus electronic shorting do to condensation of the electrical circuitry
was a major concern. To circumvent this problem we encased the laser and dual
TEC setup in an aluminum housing. Holes were drilled in this housing and air was
continuously flowed through the setup at positive pressure. The water vapour was
first removed from the air by flowing the air through a commercial dessicant called
Drierite. This setup was utilized for both the pump and probe laser arrangements.
With recent advances in DFB lasers one can now purchase lasers which are evacuated around the actual diode. These can be cooled without danger below the dew
85
point of the laboratory.
Another concern in the setup was due to the laser frequency shifting with time.
To monitor this we constructed a constant pressure Fabry Perot etalon. It consists
of an Invar tube, which has low thermal expansivity, surrounded by Nichrome
wire, along with a thermocouple to measure temperature. Two confocal mirrors
are fitted to both ends of the hollow Invar tube, and attached to a piezo. The
Nichrome wire is utilized as a heater and used via negative feedback to keep the
temperature of the Invar constant. The entire setup enclosed inside three stainless
steel tubes and evacuated. The idea was to utilize this to look for correlations
between pressure, and temperature change of the lasing frequency and stabilize
them. This is currently being tested on a different setup.
4.4.3 Miniaturization of Gyroscope
In an effort to miniaturize the current setup one runs into difficulties creating homogeneous magnetic fields. With the standard Helmholtz arrangement the active
region, where the non linearity in the field is less than 1%, is small compared to
the radius of the coils. In order to miniaturize the system a dual pair of coils were
modeled analytically. The current ratio in the two pairs was taken to be a rational
integer. Using a numerical simulation we solved for the inter coil separation as a
function of the integer current ratio while imposing elimination of 4th order nonlinearity in the field dependence at the symmetric centre of the coils. Using this
process we found that the optimal compact arrangement occurred for a turn ratio
of 1 : 1. The coils are a distance 1.2cm and 2.2cm from the location of the cell along
the field axis, and have a diameter of 3.6cm.
The analytical solution is not strictly valid in the vicinity of magnetic shielding. We used the simulation as a starting point to estimate the optimal placement
of theses coils within Mu metal shields. This was modeled using the commercial
86
Figure 4.7: Magnetic field homogeneity for a 3cm×3cm region. Here the coils are
carrying a 10mA current.
computer program Maxwell. See fig.4.7. An optimized magnetic field homogeneity was realized by simultaneously varying both the radii of the coils and their
positions independent of each other. This solution is independent to first order to
small asymmetric misplacing of the wires.
We were able to achieve a magnetic field homogeneity of 0.1% in a region 2cm
by 2cm which was produced by coils of diameter 4cm.
4.4.4 Alternate methods to heat Cell, and remove Convection noise
A source of error in the magnetometer signal is due to random walk of the laser
beam caused by the air convection used to heat the cell. In order to reduce this we
have enclosed the cell in an evacuated glass chamber. The cell is surrounded by
a 6 boron-nitride sheets. These are finally attached to a boron-nitride rod around
which is wrapped a coaxial mineral insulated heating cable. The entire enclosure
is cemented together utilizing aluminum nitride. A current is sent through the
coaxial cable, which is shorted at one end. The power dissipated through the cable
is used to heat the boron nitride walls, which heat the cell by conduction. The
87
entire enclosure is enclosed in a evacuated glass chamber which has been silvered
to reduce radiative loss of heat. The silver coating is scratched to prevent Johnson
noise from currents in the coating.
The co-axial coils carry a 10mA current and heat the cell via direct conduction.
Because of the coaxial arrangement the magnetic field due to the current is eliminated to first order. Boron-Nitride was chosen because it is has a large thermal
conductivity. (120 W/mK). Since it is not a metal it has limited magnetic properties. The fusing cement has a nominal conductivity of 110 W/mK. However
experiments were performed and indicate the conductivity of the cement form is
30 W/mK.
An arrangement utilizing Kapton heaters was tested to heat the cells. This system was theoretically modeled. A temperature gradient exist across the boronnitride oven which can be explained by heat loss due to radiation. To eliminate
this problem we are attempting to redesign the boron nitride oven, to keep the
temperature gradient to a minimum. We also theoretically studied the radiation
exchange problem between the oven and magnetic shielding. We operate such that
the magnetic shielding is not raised above its Curie temperature. It was also shown
that one could simply use a twisted pair of heater coils to heat the cell. This is the
scheme which is currently being used. The advantage over Kapton heaters is that
the twisted pair can operate at higher temperature, whereas the Kapton heaters
melt at 200C◦ . This is because the twisted wire contains a high temperature glass
sheath.
To reduce the noise further we oscillate the current in the twisted wire at a
frequency higher than the bandwidth of co-magnetometer. The bandwidth of the
co-magnetometer is ≈ 100Hz, and the current is oscillated at ≈ 25KHz. This also
ensures that the spin precess through a small angle during the period of current oscillation. This coaxial wire has an outer sheath of copper, and an inner sheath made
88
89
Figure 4.8: Silvered oven holding Boron-Nitride housing for Pyrex cell. These are all enclosed in a 4layer concentric Mumetal magnetic shield system.
of Everdur 655. This is a higher resistance non magnetic metal. These were chosen
to reduce the inhomogeneous DC field produced from the wires themselves. Other
wires such as Nichrome have been experimented with. The oven is placed inside
a glass encasing which is then evacuated. This is then silvered to reduce heat loss
due to radiation leakage. The silver surface is scratched so as to reduce the effect
of eddy currents.
A version of the gyroscope where the Mu-metal shields have been replaced
with a smaller ceramic ferrite shield has been implemented for use as a new test of
√
CPT violation. It has a realized noise of 1fT/ Hz, using a K-He co-magnetometer.
This is roughly 2.5 times more sensitive than the previous incarnation of the co√
magnetometer gyroscope which had a magnetic field sensitivity of 2.5fT/ Hz,
√
and 5 × 10−7 rad/s/ Hz. In the current setup if one switched to a K-Ne co√
magnetometer cell the expected sensitivity would be 5 × 10−8 rad/s/ Hz due to
the
21 Ne
gyromagnetic ratio being a factor of ≈ 10 smaller than that of 3 He. See
eq. (2.121).
90
Chapter 5
Measurement of parameters for
Polarizing Ne with K or Rb metal
Many of the applications currently using 3 He, including the co-magnetometer,
would benefit, and realize increased sensitivity by substituting 3 He with polarized
21 Ne
gas. In fact, tests of CPT violation using co-magnetometers would be greatly
improved if one utilizes polarized
21 Ne
gas (Kornack et al. (2008)). Additionally
the nuclear spin co-magnetometer gyroscope would realize an order of magnitude
gain in sensitivity (Kornack et al. (2005a)).
Very little is known about parameters which govern the spin-exchange polarization of
21 Ne
21 Ne.
In order to realize these applications, and test the feasibility of a
co-magnetometer the interaction parameters of 21 Ne with alkali metals must
be measured. The spin-exchange cross section σse , and magnetic field enhancement factor κ0 have only been theoretically calculated (Walker (1989a)). Furthermore there are no quantitative predictions of the neon-neon quadrupole relaxation
rate Γquad . In this work we investigate polarizing
21 Ne
with optical pumping via
spin exchange collisions and measure the relevant spin exchange rate coefficient,
magnetic field enhancement factor, and quadrupolar relaxation coefficient. Fur-
91
thermore we measure the spin destruction cross section of Rb, and K with
21 Ne,
and find agreement with the values found in the literature. Finally we discuss the
feasibility of utilizing polarized 21 Ne for operation in a co-magnetometer.
5.1
Theory
In this work we refer to the spin exchange rate measurements of the K-21 Ne system.
However the measurement technique is identical to that utilized for the Rb-21 Ne
system.
21 Ne becomes polarized via spin-exchange collisions with polarized alkali atoms.
During spin exchange collisions the electron wavefunction of the alkali atom overlaps with the noble gas nuclei. They interact via a hyperfine Fermi contact interaction of the form
Hse = α~In · S~a
(5.1)
where ~In refers to the noble gas nuclear spin operator, and S~a the alkali electron
spin operator. During collision this interaction leads to an exchange of angular
momentum from the alkali valence electron to the noble gas nuclei. The collision
also brings both the alkali valence electron, and the noble gas nuclei into close
proximity. This causes them to both experience strong magnetic fields due to the
the interaction of their magnetic moments. This results in a change in the Larmor
precession frequency of both species. This frequency shift is described in terms of a
magnetic field enhancement factor κ0 (Walker & Happer (1997)). It is defined as the
ratio of the Larmor frequency shift of each species caused by the contact interaction
and to the shift caused by the classical magnetic field generated by each other gas
species. The value of κ0 is often much greater than unity. It can be measured by
comparing the precessional frequency shift of an alkali atom the noble gas is in
contact with, to the actual magnitude of the magnetic field produced by the noble
92
gas nuclei.
For light alkali atoms the dominant spin exchange mechanism for polarizing
noble gas involves binary collisions. Here the spin-exchange rate constant can be
expressed in terms of the familiar spin-exchange cross-section as Grover (1983):
κ a = σse vK − Ne
(5.2)
Where vK − Ne is the relative velocity between the potassium, and neon atoms.
The polarization of neon atoms via spin exchange collisions with potassium
atoms of number density [K] can be described by (Walker & Happer (1997),Appelt
et al. (1998)):
3
3 ∂PNe
1
= κ a [K ](ǫ(W, β) PK − PNe ) − PNe Γquad
2 ∂t
2
2
(5.3)
Here the coefficient κ a represents the spin-exchange rate constant. The terms PK
and PNe are related to the longitudinal spin polarization of the alkali atom of spin S,
and neon atom with nuclear spin K by PNe = hWz i /W, and PK = 2 hSz i. The neon
relaxation is dominated by the quadrupolar relaxation rate Γquad (Adrian (1965)).
One method to measure κ a , suggested by Gentile (Gentile & McKeown (1993)),
is to simply measure the rate of rise of neon polarization at PNe = 0. This transforms eq. (5.3) to
∂PNe
5
= κ a [K ] PK
∂t
3
(5.4)
We measure the buildup of neon polarization as a function of time extrapolated to
the slope at zero neon polarization to determine the alkali-neon spin exchange rate
κ a . These measurements are made for times where the buildup of neon polarization remains linear.
In this work we additionally use a method based on re-polarization in the dark
93
to measure κ a (Chann & Walker (2002),Kadlecek et al. (1998)). Here the polarized
neon spins repolarize the K spins in the absence of optical pumping. In this case
the total K spin, F = I + S, evolves as
∂Fz
= D ∇2 Fz − ΓK Sz + κ a [ Ne](Wz − ǫ(W, β)Sz )
∂t
(5.5)
Here the first term represents the contribution due to diffusion of K through the
cell. The second term represents depolarization via spin destruction collisions. We
expect the contributions from diffusion to be small compared to the other terms.
The diffusion term is ≈ 2840 times smaller than the spin destruction rates. When
the neon has reached steady state polarization one can simplify eq. (5.5) as:
κa =
1
2 Γ K PK0
( 32 PNe0 − 21 ǫPK0 )[ Ne]
(5.6)
Each of these quantities can be measured independently. ΓK can be measured by
chopping the pump beam, and analyzing the resulting decay in the alkali polarization. [ Ne] is directly measured as the vapour cells are filled. P Ne0 can be computed
directly by performing NMR on the polarized sample. PK0 can be measured by
scaling the polarization when the pump beam illuminates the cell according to the
optical rotation signal in the light, and the dark. The individual measurements for
determining the spin exchange rate, spin destruction cross-sections, and κ0 values
are described in greater detail in the following sections.
The dominant source of relaxation in neon arises from the quadrupolar relaxation rate Γquad . According to Adrian (1965) the relaxation rate is proportional to
the neon filling pressure. One can measure this contribution by measuring the T1
of neon cells with different filling pressure.
The dominant source of alkali relaxation is due to spin destruction collisions.
94
In the absence of optical pumping the spin destruction rate can be expressed as:
Rsd = nK σK −K V K −K + n Ne σK − Ne V K −K + n N2 σK − N2 V K − N2
(5.7)
In order to measure σK − Ne , one measures the decay of alkali polarization to determine the total spin destruction rate and fit to eq. (5.7). One must modify the
measured relaxation rate by accounting for the paramagnetic slowing factor. This
is described in further detail in the section regarding measurement of the alkali
polarization decay.
5.2
The
Experimental Setup
21 Ne
gas sample is contained in a cubic pyrex cell of side length 20mm. It is
fully illuminated by a high power Sacher littrow diode laser which is tuned to the
K D1 line and outputs 100mW of power. This beam passes through a λ/4 waveplate before illuminating and polarizing the K atoms. A Toptica Dl-100 diode laser
is utilized as a probe beam. It passes through a linear polarizer and propagates
parallel to the pump beam. The pyrex cell contains a K droplet, ≈ 100 torr of nitrogen for quenching and 6.2 atm of 90% isotopically enriched21 Ne gas. It is heated
to 180C◦ via a hot air line which heats a glass oven. The oven is placed in the
middle of two large 34 inch diameter Helmholtz coils which create a holding field
of approximately 16.7 gauss. The Helmholtz coils are powered by a 100V Bipolar
amplifier. Perpendicular to these coils are a pair of NMR tip coils, also placed in a
Helmholtz configuration, and a pair of RF modulation coils. Along the y axis, out
of the plane of the optical table, (see fig 5.1) a NMR pickup coil is placed. It has a
Q of ≈ 14, and is tuned to the neon resonant frequency. For the Rb measurements
the pump laser is a 2W Coherent 19 diode laser array, and the probe is a 10mW
Nanoplus DFB diode.
95
Compensating Coil
z
x
Feedback
y
Fluxgate
Polarizing
Beamsplitter
NMR Pick-up
Oven
Probe Laser
Cell
Photodiodes
Subtraction
Linear
Polarizer
NMR tip Coils
Current
Source
λ/4
Lock-In
Amplifier
(Circular Polarizer)
Pump Laser
Modulation
Reference
(Larmor Frequency)
Feedback
AFP frequency
Modulation
AFP Amplitude
Modulation
Magnetometer Signal
Figure 5.1: Experimental Setup
κ0 is measured with a sequence of EPR-AFP flips, followed by NMR tips. κ0 is
calculated by computing the ratio of the magnetic field shift experienced by the K
(EPR-AFP) and the magnetic field produced by the neon, as determined from the
NMR.
To determine the spin exchange constant using the repolarization method a
series of EPR-AFP flips was used to determine the K frequency shift. PNe0 was
calculated using the measured value for κ0 and the neon pressure in the cell which
was measured during filling. [ Ne] is known from the cell filling pressure.
To determine PK0 a two step process was used. First the RF source was swept
over the K Zeeman levels to determine the K polarization with the pump on. The
pump beam was then blocked and the repolarization of K by Ne was measured by
monitoring the optical rotation. PK0 was calculated by calibrating the optical rotation obtained during back polarization with the optical rotation measured while
the cell was illuminated. The zero optical rotation point when the pump beam is
blocked was found by flipping the neon spins with AFP and monitoring the subsequent change in the K optical rotation in the dark. ΓK is determined by manually
96
chopping the pump beam, and measuring the resulting decay in the optical rotation.
For the rate of rise measurements the average alkali polarization in the cell PK
must be measured while the cell is illuminated. To account for the variation in
K polarization across the cell, the probe beam was swept across the cell and the
optical rotation was measured. At the centre of the cell RF modulation was used
to measure the K polarization. By scaling the polarization as compared to the
optical rotation across the cell an estimate of the average polarization of the cell
was obtained.
By comparing the T1 data from the manufactured cells, and those constructed
by others the contribution of neon-neon quadrupolar collisions to overall neon
spin relaxation can be estimated. The T1 times were calculated by utilizing EPR
lock-AFP flips to calculate the neon polarization.
A measurement of the spin destruction rate of rubidium in the rubidium-neon
cell and subsequent calculation of the Rb-Ne spin destruction cross section was
carried out. The spin destruction rate was measured via the rubidium relaxation
in the dark measurement which was previously described. The spin destruction
rate was extrapolated to zero probe beam intensity by making multiple spin destruction rate measurements at different probe beam intensities. The variation in
probe beam intensity was achieved by placing neutral density filters of different
values directly in the beam path before the probe entered the cell.
In the following subsections we describe each of the individual measurements
in greater detail.
5.2.1 NMR detection
In order to determine the spin-exchange rate constant, several different quantities for each rate constant must be measured. P Ne0 is measured using NMR. To
97
accomplish this the neon NMR was obtained by tipping the
21 Ne
using two cal-
ibrated 9 inch diameter coils in the Helmholtz configuration. Typical tip angles
are approximately 30◦ . The resulting neon NMR is detected with a 4 coil pickup
with 780 turns per coil. This consists of two pairs of pickup coils with opposite
winding orientation so that the combined pickup has zero dipole moment. This is
done to minimize coupling of the pickup coil to external fields which are not due
to the
21 Ne
gas. Each pair of coils are located symmetrically about the cell, and
are wound on one Teflon rod. Each coil has 780 turns of enameled copper magnet
wire. A numerical model was written to determine optimum coil placement in
order to maximize pickup sensitivity.
In order to minimize systematic errors the tip coils were calibrated using two
independent methods. First the pyrex cell is replaced with a small pickup coil of
similiar dimension to the pyrex cell, while the tip coils are operated. The resulting signal is fed into a lockin amplifier and used to calibrate the magnetic field
strength. Care is taken to properly align the axis of the pickup coil with that of
the tipping coils. Second, the current entering the tipping coils is measured with
a clamp on ammeter. The magnetic field produced from the tipping coils is calculated from a knowledge of this current and the coil separation, and diameter.
The two calibration schemes agree to 2%, which is within the compound precision of the coil geometry measurements, alignment of pickup coils, and ammeter
precision.
For NMR detection the pickup signal is fed into a low noise pre-amp, and monitored via computer. A typical neon NMR signal is shown in fig.5.2. The pickup coil
is also calibrated using two methods. The first requires a dummy source of known
size to create a magnetic field. The magnetic field strength is calculated theoretically based on the dummy coil geometry. The resulting pickup is measured to
produce a calibration. The pickup coils are also operated in reverse to produce a
98
0.60.6
0.4
0.4
0.2
0.2
NMR
Pickup
0.0
gain 1e4
(volts)
0.
-0.2
0.2
-0.4
0.4
-0.6
0.6
-0.8
0.8
0
0.05
0.10
0.15
0.20
Time (s)
Figure 5.2: Representative NMR signal of polarized 21 Ne gas
magnetic field at the cell location. A small pickup coil is placed here. Using reciprocity arguments (Jackson (1999)) the two signals are compared and agree to 6%.
The uncertainty of the Q of the resonant coils is 4%.
5.2.2 Electron Paramagnetic Resonance Shift
In order to calculate the κ0 value, the effective magnetic field the K atoms experience must be measured simultaneously with the NMR signal. This effective magnetic field is measured by monitoring the shift in the frequency in the electron
paramagnetic resonance(EPR)upon reversal of the neon spins. (Romalis & Cates
(1998),Schaefer et al. (1989b),Newbury et al. (1993),Barton et al. (1994)). A double
feedback scheme is utilized to accomplish this. The first feedback loop locks the
output of the holding field to minimize magnetic field fluctuations from external
sources. To accomplish this a fluxgate magnetometer is placed approximately 5
inches from the pyrex cell outside the glass oven. In order to prevent the fluxgate
from saturating, from the 16.7gauss field produced from the large 34 inch diameter
Helmholtz coils, a solenoid was wound around the fluxgate. This compensating
solenoid produces a field which cancels the holding field and results in zero field
at the fluxgate. The compensating solenoid is powered by a custom built voltage
99
controlled current supply. The output of the fluxgate is fed through PID feedback
electronics to the input of the voltage controlled power supply which powers the
large Helmholtz coil holding field.
The second feedback loop locks the magnetic holding field to the K resonance.
Thus when the K resonance is shifted the holding field for the atoms is also shifted
so that the K resonance frequency remains constant. In short the two feedback
loops are summed. The first feedback controls the current which supply the holding field for the atoms, while the second feedback give an offset to this field in
order to maintain a constant K precession frequency.
In order to operate the second feedback loop a small RF coil is used to produce
a magnetic field at 12.4957MHz, with a sweep width of 15KHz, and a sweep rate
of 340Hz. The optical rotation of the probe beam is measured as the RF coils sweep
in frequency across the K magnetic resonance. The derivative of this signal is produced taking the out of phase component after feeding the signal into the lockin
amplifier which has been referenced to the RF sweep rate. The out of phase component is fed into an integral feedback box and used to control the voltage controlled
current source which powers the compensating solenoid. In this manner we can
lock to the K resonance.
The output of the integral feedback box is monitored and calibrated by introducing known frequency shifts in the base RF field. That is the RF field frequency
is shifted by 100Hz, and the resulting feedback box voltage is monitored. This relation is linear, and is used to calibrate the EPR frequency shifts. This technique
is utilized to lock to the end resonance of the F = 2 manifold of the K spectrum.
Operating at high field causes the K resonance to be non-linearly spaced due to
the Breit-Rabi splitting. The neon NMR frequency is approximately 5500Hz while
the system is locked to the K end state. A numerical program is used to calculate the effective gyromagnetic ratio of the end state due to the Breit-Rabi splitting
100
400
Frequency Shift (Hz)
200
0
-200
-400
-600
0
5
10
15
20
25
Time (s)
Figure 5.3: Representative EPR shifts after 2 hours of polarization. The spikes in
the data are due to temporary loss of lock while the AFP coils are operated.
(Woodgate (1989)). This coupled with the NMR data allows the determination of
κ0 (Romalis & Cates (1998)).
∆ν =
8π dν( F, M )
κ0 µK [ Ne] PNe
3
dB
(5.8)
In order to measure the magnetic field enhancement factor κ0 the technique
of Adiabatic fast passage (Abragam (1961)) is used to flip the orientation of the
polarized neon spins by 180◦ , while keeping locked to the EPR frequency and
monitoring the frequency shift. See fig5.3. The peak to peak amplitude of the
frequency shift seen using EPR is twice the value of the frequency shift induced by
the polarized neon. In order to flip the spin orientation a magnetic field is swept
in frequency across the neon NMR resonance. This is accomplished by producing a field with the small set of Helmholtz coils while satisfying the adiabatic fast
passage (AFP)conditions (Abragam (1961)):
γω̇
k∇ Bz k2
<<
<< γB1
2
B1
B1
101
(5.9)
To sweep the frequency of the AFP flipping field a voltage controlled oscillator
is fed through a multiplier circuit, and a ramping voltage from a NI-Daq analog
output. The resulting signal is then amplified through a 100V Bipolar amplifier to
supply the AFP field. The neon spins are flipped every 5 seconds. The frequency is
swept from 2500Hz to 7500Hz in 4s. The AFP field strength is ≈ 4Gauss. The flip
efficiency is 99.7% In order to optimize flip efficiency it was necessary to slowly
ramp up the AFP field strength before the AFP field frequency was swept. The
AFP field was raised from 0 to 4 Gauss in 0.7s.
5.2.3 Alkali Polarization
To determine the alkali polarization when the cell is fully illuminated, a RF field
is swept across the K Zeeman levels, while the optical rotation of the longitudinal
probe beam is measured (Chann & Walker (2002)). The transverse RF field causes
a depopulation of the end state, and a lower polarization. This in turn reduces
the optical rotation of the probe laser. By comparing the area under the peaks of
the different transitions of the F = 2 manifold and using eq. (5.11) one is able to
determine the K polarization. See fig5.4. The area under a transition peak ( F, m) →
( F, m − 1) depends on the state population ρ Fm , and the RF field Br f as (Chann &
Walker (2002)):
A Fm ∝ Br2f [ F ( F + 1) − m(m − 1)](ρ Fm − ρ Fm−1 )
(5.10)
The spin exchange rate measurement experiment operates in the regime where
the magnetic sublevel populations can be described by a spin temperature distribution (Walker & Happer (1997)). Substituting the spin temperature condition
ρ Fm ∝ exp( βm), and noting that the spin polarization is given by PK = tanh( β/2)
results in
102
Optical Rotation (arb. units)
4
3
2
1
0
-1
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Time (s)
Figure 5.4: Sweep over the A22 , A21 and A20 transitions at reduced Pump beam
power.
PK =
7r − 3
7r + 3
(5.11)
where
r=
A22
A21 + A11
(5.12)
where the state is designated with the indices AFm . Under normal operating conditions the spin polarization is very close to 1. Only the end state transition A22 is
visible under these conditions. However one can view the entire spectrum if the
pump beam power is sufficiently attenuated.
5.2.4 Alkali Polarization Decay Constant measurement
In order to measure the K decay constant the optical rotation due to the K is monitored, as the pump beam is manually chopped. See fig5.5. The decay constant is
then determined by fitting the decay curve on a log plot. One must account for
the fact that to obtain the true time constant the measured constant must be multiplied by the paramagnetic coefficient or ’slowing down factor’ (Walker & Happer
(1997),Chann & Walker (2002),Appelt et al. (1998)).
103
1.0
Absolute K Polarization
0.8
0.6
0.4
0.2
0.0
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Time (s)
Figure 5.5: Potassium Polarization decays as a result of Pump beam being manually chopped.
For spin K, and spin temperature β the slowing down factor is given by eq.
(2.21): For β << 1 this reduces to 4I ( I + 1)/3 see eq. (2.24)(Walker & Happer
(1997)). Only decay data from the low polarization portion of the data set were fit
to the β << 1 simplified expression.
5.2.5 Back Polarization measurement
In order to measure the K polarization in the dark, the optical rotation due to Ne-K
back polarization was measured. The optical rotation signal is routed through a
low noise Pre-amp and fed into a NI-DAQ for acquisition. To measure the zero
polarization level the AFP Helmholtz coils are run to flip the neon polarization.
See fig5.6.
To determine the back polarization in the dark the gain adjusted optical rotation to the optical rotation signal with the pump beam is illuminating the cell is
compared to that when the pump beam is blocked. Since the alkali polarization
was measured when the pump beam was unblocked one can scale the optical rotation signals to the polarization. During operation of the pump beam large (≈ 1)
radian optical rotations were observed. In this regime one can no longer use the
104
6x10
-4
K Back Polarization
4
2
0
-2
-4
-6
0
5
10
15
20
25
Time (s)
Figure 5.6: Absolute Potassium Back polarization as Neon is flipped via Adiabatic
fast passage
standard small angle optical rotation formula
Φ=
I1 − I2
2( I1 + I2 )
(5.13)
Where I1 and I2 are the voltages on the individual channels of the balanced polarimeter. Instead one can use:
1
Φ = sin−1
2
I1 − I2
I1 + I2
(5.14)
5.2.6 Alkali density measurement
In order to calculate κ a using the rate of rise method, the alkali density must be
independently measured. This is accomplished by measuring both the optical rotation of the longitudinal probe beam and its detuning. This enables calculation of
the alkali density, while accounting for the fact that the alkali is not polarized to
1. The detuning is measured by monitoring the optical rotation as the probe beam
frequency is swept. The probe beam frequency is tuned by varying the temperature, so as to minimize variation in the probe beam intensity. The probe beam is
105
detuned ≈ 0.5nm from the centre of the absorption peak. The pump beam is tuned
to resonance by varying the current until the optical rotation experienced by the
probe beam is a maximum. There is sufficient pump beam power, 100mW so that
the cell is uniformly polarized.
To determine the alkali density the optical rotation is measured and fit to:
1
1
π
Θ = lnre Px − Im [ L(ν − νD1 )] + Im [ L(ν − νD2 )]
2
3
3
(5.15)
where the Lorentzian line shape is given by:
L(ν − ν0 ) =
Γl /2π + i (ν − ν0 )/π
(ν − ν0 )2 + (Γl /2)2
(5.16)
Here n is the density of K in the cell, l is the path length which the probe beam
propagates through the cell, and re is the classical electron radius. The contribution to the optical rotation from the D2 transition is on the order of a few percent.
The density is 4.7 × 1013 /cm3 which is a factor of 4 less than that predicted by the
empirical formula which relates the vapour pressure to the cell temperature. However there is often a discrepancy in the value calculated from the saturated vapour
pressure and the actual vapour density by at least a factor of 2 (Chann & Walker
(2002)). This comes from the fact that the cell may not be of uniform temperature,
or from substantial heating due to the pump beam. This effect can also be caused
by absorption, or a reduced alkali vapour pressure due to interactions with the
glass walls of the cell. This has been observed for pyrex glass cells, which is the
same glass employed in these measurements.
106
5.3
Fermi Contact interaction κ0 Results
κ0 for the interaction of K with 21 Ne is 30.8 ± 2.7. This is approximately 10% lower
than the value of 34 as predicted by Walker Walker (1989b). Although the values
do not agree with the predictions it should be noted that the general trend among
the predicted values, and experimentally measured κ0 for other noble gas-alkali
pairs is for the predicted values to be 10 − 20% higher than those obtained experimentally. When the experimentally measured values are arranged by magnitude
the gas mixtures show the same order as those of the theoretically predicted values. In a Rb-Ne system Walsworth has measured κ0 to be 32.0 ± 2.9 (Stoner &
Walsworth (2002)), while the value predicted by Walker is 38 (Walker (1989b)).
The Fermi-contact interaction was also measured for the rubidium-neon pair.
It was found to be 35.7 ± 3.7 which is in agreement with the value found by Stoner
& Walsworth (2002) of 32.0 ± 2.9. Both of these values are below that predicted
by Walker (1989a) of 38. However both of the experimental measurements follow the expected trend of being larger than the Fermi-contact interaction for the
potassium-neon pair. See Table 5.3
5.4
Results of neon quadrupolar relaxation Γquad measurement
By comparing the T1 data from the manufactured cells, and those constructed by
others the contribution of neon-neon quadrupolar collisions to overall neon spin
relaxation can be estimated. In general one expects that for noble gases with a
quadrupole moment, the dominant form of relaxation is due to nuclear electric
quadrupole interaction (Adrian (1965)). As a consequence of this one expects the
spin-lattice relaxation time to vary inversely with the density of the buffer gas. In
107
Quadrupolar Relaxation due to Neon Collisions
0.035
1/Lifetime (1/min)
0.03
0.025
0.02
0.015
0.01
0.005
0
0
1
2
3
4
5
6
7
Cell Pressure (Amagat )
Figure 5.7: Neon relaxation as a function of cell pressure. Here the data satisfies
the relationship Pressure×T1 = 214 ± 10Amagat×min. The fact that the T1 is inversely proportional to the cell filling pressure seems to indicate that the dominant
relaxation mechanism is due to nuclear electric quadrupole collisions. This is consistent with what one would expect from a buffer gas with spin greater than 1/2.
fig 5.7 we see that the cells follow this linear relationship. The data from the cell
with the lowest pressure is from Grover (Grover (1983)) whereas the data from the
other cells is from this work. Both Grover’s cell and 3.34 atm cell is filled with a
Rb-Ne mixture, whereas the others are filled with a K-Ne mixture. The pressure
in the Rb-Ne cell is determined by measuring the broadening of the optical D1
transition in Rb. The reason for the large uncertainty in the 3.34 atm cell’s pressure
is due to the uncertainty in the literature of the broadening parameters for the D1
transition of Rb in neon gas (Ottinger et al. (1975)).
5.5
Spin exchange Rate coefficient Results
The K-21 Ne spin exchange rate, measured with the repolarization method, is 3.36 ±
0.42 × 10−20 cm3 /s. The value obtained by using the theoretically predicted spin
exchange cross section by Walker is 7.5 × 10−20 cm3 /s. See Table5.1.
Data for the rate of rise method was taken in the limit of low neon polarization.
108
7x10
-4
Neon Polarization
6
5
4
3
2
1
0
0
1
2
3
4
5
Time (minutes)
Figure 5.8: Absolute neon polarization as function of time to determine spin exchange rate constant by rate of rise method.
Here the rate of rise of neon polarization remained linear. See fig5.8. The rate of
rise method gives a value of 2.34 ± 0.29 × 10−20 cm3 /s for the K-21 Ne spin exchange
rate.
Additionally neon was polarized using spin exchange optical pumping with Rb
metal. The Rubidium-Neon spin exchange rate was also measured and found to be
0.80 ± 0.12 × 10−19 cm3 /s. Walker (1989a) predicts a value of 1.66 × 10−19 cm3 /s.
See Table 5.2. However this value is not in agreement with the previously measured value by Chupp & Coulter (1985) of 4.66 × 10−19 cm3 /s. However Chupp
& Coulter (1985) determined their rubidium density by directly applying the saturated vapour pressure calculated by the empirical formula given by Alcock et al.
(1984). This formula can be in disagreement with the actual alkali density by as
much as a factor of two (Chann & Walker (2002)). Chupp fits his data directly
through the origin implying that the contribution of neon quadrupolar relaxation
collisions to the T1 is negligible. We have shown in this work that this is incorrect,
and that the dominant contribution to the neon T1 is in fact due to quadrupolar
collisions. This coupled with the fact that the more precise measurements carried
out by Chann & Walker (2002) indicate a statistical variation in the rate constants of
109
Spin Exchange Rate Method
Repolarization
Theoretical Prediction
Rate of Rise
cm3 /s
3.36 ± 0.42 × 10−20
7.5 × 10−20
2.34 ± 0.29 × 10−20 cm2
Table 5.1: Spin exchange parameter of a potassium-neon system
Spin Exchange Rate Method
Repolarization
Theoretical Prediction
Rate of Rise
cm3 /s
0.80 ± 0.12 × 10−19
1.66 × 10−19
0.82 ± 0.18 × 10−19
Table 5.2: Spin exchange parameter of a rubidium-neon system
25% upon subsequent measurements implies that this disagreement is not entirely
unexpected.
5.6
Measurement of Spin destruction cross-sections of
neon with Rb and K
Utilizing the data from the potassium relaxation in the dark measurement one can
compare the measured spin destruction rate to that predicted by using the known
Species
K-Ne (this work)
K-Ne (prediction) Walker (1989a)
Rb-Ne (this work)
Rb-Ne (Walsworth) Stoner & Walsworth (2002)
Rb-Ne (prediction) Walker (1989a)
Fermi Contact Interaction
30.8 ± 2.7
34
35.7 ± 3.7
32.0 ± 2.7
38
Table 5.3: Fermi contact interaction measurements for the K-Ne, and Rb-Ne systems in this work, and compared to both predictions ans measurements by other
groups.
110
4.0x10
-20
Spin Exchange Rate (cm3/s)
3.8
3.6
3.4
3.2
3.0
2.8
2.6
0
1
2
3
4
5
6
Figure 5.9: Scatter in Spin exchange rate measurements for K-Ne
1.0x10
-19
Spin Exchange Rate (cm3/s)
0.9
0.8
0.7
0.6
0.5
0
1
2
3
4
5
Figure 5.10: Scatter in Spin exchange rate measurements for Rb-Ne
111
Species
K-Ne
K-Ne
K-Ne
Rb-Ne
Rb-Ne
Rb-Ne
Rb-Ne
Group
This work
Franz Franz & Volk (1982)
Walker (prediction) Walker (1989a)
This work
Franz and Volk Franz & Volk (1976)
Franzen Franzen (1959)
Walker (prediction) Walker (1989a)
10−23 cm2
1.1 ± 0.1
1.41 ± 0.14
1.6
1.9 ± 0.2
1.9
5.2
1.8
Table 5.4: Spin destruction cross sections for K-Ne, and Rb-Ne as compared to both
theory and other measurements
spin destruction cross sections, and gas densities in the cells using eq. (5.7).
For the case of potassium with neon the predicted spin destruction rate is 8%
smaller in the 1.6atm cell, and 8%greater in the 6.2atm cell than the measured
value. This is reasonable agreement since the relevant spin destruction cross sections have only been measured to one significant figure. The spin destruction rates
are approximately 2840 times larger than the contribution due to gradient relaxation. It is larger than the rate of diffusion to the cell wall by a factor of 40.
A measurement of the spin destruction rate of rubidium in the rubidium-neon
cell and subsequent calculation of the Rb-Ne spin destruction cross section was
carried out. The spin destruction rate was measured via the rubidium relaxation
in the dark measurement which was previously described. The spin destruction
rate was extrapolated to zero probe beam intensity by making multiple spin destruction rate measurements at different probe beam intensities. The variation in
probe beam intensity was achieved by placing neutral density filters of different
values directly in the beam path before the probe entered the cell. The alkali-Ne
spin destruction cross sections are listed in Table 5.4.
112
One can accurately predict the polarization of the neon gas by using eq. (2.105).
PNe = Palkali
Rse
ǫ
W/S Rse + 1/T1
(5.17)
Here the measured spin exchange rate for the Rse was substituted into eq. (5.17)
and T1 was calculated assuming that the quadrupolar relaxation mechanism is the
dominant relaxation mechanism. Making these assumptions, while taking into
account the non-uniform alkali polarization profile across the cell,a polarization of
0.55% at 140◦ C was expected. A value of 0.8 ± 0.13% was measured using EPR.
5.7
Conclusion
The spin exchange rate coefficient for both the Rb-21 Ne, and K-21 Ne systems have
been measured using two different techniques. The values from the rate of rise
and repolarization techniques are consistent. The spin exchange rate coefficient
for Rb-21 Ne does not agree with the previous value in the literature. We claim this
discrepency is caused by the exclusion of relaxation due to the neon quadrupolar
relaxation from previous measurements. The Fermi contact interaction for both
the Rb-21 Ne, and K-21 Ne pairs have been measured. There is agreement with the
previously measured value for the Rb-21 Ne system. The neon quadrupolar relaxation has been measured. Also the various alkali spins destruction cross-sections
with
21 Ne
have been measured. These agree with the previous values quoted in
the literature. We have modeled the neon polarization as a function of alkali density and have show that the polarization dynamics are well described by assuming
quadrupolar relaxation is the dominant form of relaxation, and spin exchange the
dominant polarizing interaction.
113
Chapter 6
Feasibility of utilizing 21Ne in a
co-magnetometer
The main objective of these experiments on 21 Ne is to ultimately create a neon comagnetometer which can be used for experiments on tests of fundamental symmetries, and for deployment as a sensitive gyroscope. However application of
21 Ne
in a co-magnetometer requires polarization higher than that observed for the set
of spin-exchange rate measurements at 140◦ C.
One can increase the neon polarization by increasing the alkali density. However this requires additional laser power to ensure the optically thick cell remains
uniformly polarized. Experiments were carried out at higher density at 180◦ C and
resulted in ≈ 8% neon polarization.
A Mathematica model was used to calculate the sensitivity of the co-magnetometer
at normal operating temperatures by optimizing the laser power. Here the sources
of noise included were the effects of spin projection noise, and photon shot noise.
The first effect is due to the Heisenberg uncertainty principle as applied to the
transverse components of the spins and the fact that they do not commute. The
second source of noise is the noise of the rotation signal of the probe beam for a
114
balanced polarimeter setup. We ignore the effects of light shift noise because we
presume to operate with the pump and probe beams orthogonal to each other. In
this orientation the magnetometer signal is sensitive to the axis perpendicular to
the plane which contains both laser beams. Thus it is in-sensitive to light shift
noise caused by fluctuations in the ellipticity of the probe beam to first order. Let
us describe the Rb-Ne comagnetometer simulation in greater depth.
6.1
Effects of Light Propogation and alkali relaxation
on Rb-Ne co-magnetometer simulation
The propagation of the pump and probe beams must be modeled as the beams can
be strongly absorbed and result in non-uniform polarization through the cell. The
propagation of light through the cell is a function of the alkali polarization in the
cell which is given by eq. (2.18). The propagation dynamics can be described by:
dRop
= −nσ(ν0 )(1 − Pequil ) Rop
dx
As one can see the attenuation
dRop
dx
(6.1)
vanishes if the alkali is fully polarized. How-
ever this is never achieved in practice due to spin relaxation due to spin destruction
collisions. These can be described by:
sd
se
sd
Rrel = n Rb σRb
− Rb V Rb + n Ne σRb− Ne V Ne + R pr + ( ǫ + 1) Rwall + R Rb− Ne n Rb
(6.2)
Rwall is given by eq. (2.62), V i− j is given by eq. (2.39), ni refers to the density per
cm−3 of species i, and ǫ is given by eq. (2.56). R pr is the pumping rate due to the
115
probe beam and is given by:
λ probe
×
σRb− xs (λ probe ) × Pprobe
wlhc
1
Exp −σRb− xs (λ probe )n Rb
2
R pr =
(6.3)
w refers to the width of the cell, and l its length. h is Planck’s constant, c is the
speed of light, and P probe is the probe beam power. The photon absorption cross
section σRb− xs is given by:
σRb− xs (ν) = re c ( f D1 V(ν − νD1 ) + f D2 V(ν − νD2 ))
(6.4)
Solution of eq. (6.1) gives a pumping rate profile across the cell as:

Rop ( x ) = Rrel W  Rop−int
Exp[
Rop−int
Rrel
− σRb−xs (λ D1 )n Rb x ]
Rrel


(6.5)
Here P pump is the pump beam power, and x is the propagation distance through
the cell. The function W is the principal value of the Lambert W-function. This is
defined as the inverse of the function f (W ) = WeW . It is also refer
Rop−int = σRb− xs (λ D1 ) Ppump λ D1
1
Ahc
(6.6)
where A is the cross sectional area of the cell, and λ D1 is the wavelength of the D1
line.
An alkali polarization profile for a pancake cell of dimensions 6 × 15mm under
condition of high pumping rate (≈ 7800s−1 ) is shown in fig.6.1
116
Absolute Polarization
0.98
0.96
0.94
0.92
0.1
0.2
0.3
0.4
0.5
0.6
Distance Across cell (cm)
Figure 6.1: Absolute Rb polarization as function of propagation distance through
cell, for pancake cell of dimensions 6 × 15 × 15mm and a pumping rate of ≈
7800s−1 .
6.2
Simulation of Noble gas relaxation
We are able to relate the noble gas polarization in the steady state to the alkali polarization using eq. (2.105). However eq. (2.105) indicates that the steady state
polarization of the noble gas not equilibrate with the alkali polarization due to
strong noble gas relaxation mechanisms. In this work we model the effects of both
quadrupolar relaxation, and relaxation due to diffusion of the noble gas in a magnetic field gradient.
The quadrupolar relaxation rate of neon is taken from the measured relaxation
rate as a function of cell pressure. This is depicted in fig.5.7.
The magnetic field gradient in the cell can be evaluated utilizing the technique
of magnetic vector potential(Jackson (1999)). The field inside the cell can be modeled as a uniformly polarized mass. The magnetic field at an arbitrary point inside
the field can be evaluated by replacing the polarized mass with a surface current
of density equal to the cell’s magnetization (Jackson (1999)). This enables calculation of the relaxation rate given by eq. (2.64) on a point by point basis, which is
averaged over the cell. This is calculated numerically on a lattice. The noble gas T1
117
then becomes:
Pneon
1
=
+
T1
12840s
Z
volume
D Ne− Ne
∇ B( Pneon )
B( Pneon )
2
dV
(6.7)
In the above equation Pneon represents the pressure of neon in the cell in amagat. The denominator of the first term describes the relaxation of neon due to
quadrupolar relaxation. The
Pneon
12840
is the quadrupolar relaxation rate given in s−1 .
It is equal to 214Amagat·mins. D Ne− Ne is the self diffusion coefficient of neon, and
∇ B is the magnetic field gradient taken at every point in the cell. The eq. (6.7)
gives
1
T1
in units of s−1 . For a cell with pressure of 3 amagat the relaxation rate due
to diffusion is ≈ 30% as large as the quadrupolar relaxation rate.
6.3
Noise mechanisms in a Rb-Ne co-magnetometer
We describe the noise contributions to the magnetometer sensitivity in more detail.
The total noise in the magnetic field measurement has two major contributions.
The first is the spin projection noise due to the quantum uncertainty in the spin’s
orientation. The second contribution is due to the photon shot noise. To obtain the
total noise in the magnetic field measurement these contributions must be added
in quadrature.
δB =
q
2 + δB2
δBspn
psn
(6.8)
The descriptions of the uncertainty mechanisms below follow that of Savukov et al.
(2005).
118
6.3.1 Spin Projection Noise
To determine the spin projection noise of the co-magnetometer let us first consider
the commutation relation between the transverse components of the alkali spin.
[ Fx , Fy ] = iFz
(6.9)
This leads to the uncertainty relation
δFx δFy ≥
| Fz |
2
(6.10)
Let us operate under conditions of full polarization since this minimizes the uncertainty relation. Due to symmetry δFx = δFy for spins which are not in a squeezed
state. We can describe the signal from N alkali atoms as making N uncorrelated
measurements. In such a case the uncertainty in the spin of the transverse components becomes:
δFx =
r
h Fz i
2N
(6.11)
It is important to note that only uncorrelated measurements improve the sensitivity of the measurement. In the atomic magnetometer setup the probe beam continuously measures the spin projection. Correlated measurements from the same
atoms do not improve sensitivity. To take this into consideration let us define a
quantity χ(τ ) as the degree of loss of spin coherence from a measurement at time
t = 0, and the time t = τ.
χ(τ ) = e−τ/T2
(6.12)
Gardner (1990) gives the total uncertainty in a continuous measurement as:
1/2
Z t
τ
2
1−
χ(τ )dτ
δ h Fx i = δFx
τ 0
t
119
(6.13)
"
δ h Fx i = δFx
2T2 2T22 (e−t/T2 − 1)
+
τ
t2
#1/2
(6.14)
Since typical measurement time for the magnetometer system satisfies t >> T2 we
can combine eq. (6.11), and eq. (6.14) to obtain the total uncertainty:
δ h Fx i =
r
2Fz T2 BW
N
(6.15)
where BW= 1/2t is the bandwidth of the measurement. Here N describes the
total number of alkali spin that the probe beam interacts with. In units of root
mean square noise per root Hz we can rewrite the total uncertainty as:
δ h Fx irms =
r
2Fz T2
N
(6.16)
This can be written in terms of magnetic noise by using eq. (2.35) and by setting
S → Fx /2 and Pze → Fz /2 to find:
δ h Birms =
δ h Fx i Rtot
γe Fz
(6.17)
6.3.2 Photon Shot Noise
We will describe the photon shot noise associated with detection of a probe beam
with a balanced polarimeter setup. It is convenient to define the total photon flux
as:
Φ′ =
Z
ΦdA
A
(6.18)
where Φ is the photon flux per unit area, and Φ is the total photon flux taken over
the area of the probe beam. In terms of the photon flux in each photodiode we can
write the optical rotation as:
θ=
Φ1′ − Φ2′
2(Φ1′ + Φ2′ )
120
(6.19)
If we assume the rotation angle θ is small we can state Φ1′ ≈ Φ2′ . In this case we
can write the fluctuation in the photon flux as:
δΦ1′
=
δΦ2′
=
r
Φ′
2
(6.20)
This leads to a fluctuation in the optical rotation angle given by:
v
"
u
2 2 #
u
δθ
δθ
δ hθ i = t2BW
+
δΦ1′
δΦ1′
δΦ1′
δΦ1′
δ hθ i =
r
BW
2Φ′
(6.21)
(6.22)
In units of root-mean square noise per root Hz this becomes:
δ hθ i =
r
1
2Φ′
(6.23)
For the magnetometer it is more useful to relate the noise in terms of the rotation
angle to the atomic polarization. Consider a probe beam detuned from the D1
resonance. We can calculate the optical rotation due to the D1 resonance if we
combine eq. (6.23) and eq. (2.81) to obtain:
δ h Px irms =
πlnre c f
√
2
Φ′ Im[V(ν − ν0 )]
(6.24)
Where Φ′ is given by:
Φ′ =
ηPprobe λ probe
Exp −nrb lσRb− xs (λ probe )
hc
121
(6.25)
Here η is the quantum efficiency
η=
( Resp)hc
λ probe e
(6.26)
where Resp is the responsivity of the photodiode in A/W, and e is the charge of the
electron. The noise in angular units can be converted to magnetic units by using
eq. (2.35) and noting that S = Re( Pxe ) to obtain:
Px
γe B0
=
Pz
Rtot
(6.27)
or,
δ h Birms =
6.4
δ h Px irms Rtot
γe Pze
(6.28)
Results of Rb-Ne co-magnetometer simulation
For a cell which is cubic with an edge of 5mm at 180◦ C with 15mW pump beam
√
one finds the sensitivity averaged over the cell to be 0.77fT/ Hz and compensation point of 4.7Hz. At 200◦ C with 15mW pump beam one finds the sensitivity
√
averaged over the cell to be 0.31fT/ Hzand compensation point of 4.7Hz. These
√
sensitivities are both below the 1fT/ Hz noise floor caused by technical sources.
Thus we could operate a co-magnetometer with these settings. In both of these
cases the probe beam was optimized to 10mW and detuned 0.39nm lower than the
resonant wavelength.
For the situation where we operate at 240◦ C we find that pumping with 75mW
√
gives a sensitivity of 0.67fT/ Hz. This is the sensitivity averaged over the entire
cell. That is with a probe beam expanded over the entire cell.
√
If instead we go to a pancake geometry one can reach 7aT/ Hz for a cell of
dimensions 6 × 15 × 15mm with 3atm of neon. This is a reasonable since the pyrex
122
glass we have available has wall thickness of 1mm, which corresponds to being
safely able to encase 3atm nominally. Also the cell temperature is 180◦ C which
is near the upper temperature a pyrex cell be operated at without discoloration.
Thus it should be safe to operate at this temperature for a pyrex cell. The neon
frequency at the compensation point is 0.8Hz. This is not high enough that the comagnetometer zero-ing routines can be operated in a reasonable time. The pump
laser power is 40mW, and the probe beam is 10mW and detuned 0.4nm below the
D1 resonance wavelength. Typically one utilizes ultra stable single frequency DFB
laser diodes for precision experiments. These DFB diodes are normally available
with power rating up to 100mW. Thus it should be easy to obtain laser for pumping
and probing with the required power rating. The angular noise corresponding
√
to the atomic shot noise is 5nRad/ Hz in this scheme. Thus it seems feasible
to create a rubidium-neon co-magnetometer with fundamental noise approaching
comparable to the state of the art atomic co-magnetometers.
We can use the experimentally measured values of the alkali-neon spin exchange rate to predict the feasibility of a alkali-neon co-magnetometer. One can
accurately predict the polarization of the neon gas by using eq. (2.105). Here we
ab and calculate the T assumsubstitute the measured spin exchange rate for the Rse
1
ing that the quadrupolar relaxation mechanism is the dominant relaxation mechanism. Making these assumptions while taking into account the non-uniform alkali
polarization profile across the cell we expect a polarization of 0.55% at 140◦ C. A
value of 0.8% which was measured using EPR.
Ultimately the quantity of interest when constructing a co-magnetometer is the
frequency of the noble gas when operating at the compensation point. This is
because the frequency of the noble gas sets the time scale for the co-magnetometer.
When processes are performed adiabatically for the co-magnetometer we really
mean that they are slow compared to the noble gas precession frequency at the
123
compensation point. Thus by increasing this frequency we are able to perform
adiabatic tasks, such as zeroing, faster. For practical purposes this is of great use
when running the co-magnetometer, and gyroscope experiments.
By predicting the noble gas polarization and utilizing the noble gas gyromagnetic ratio we can predict the noble gas precession frequency at the compensation
point. Typically for operation of the co-magnetometer the noble gas frequency
must be set to at least 7Hz. Again utilizing eq. (2.105) we calculate the frequency
at the compensation point to be 2.30Hz at 180◦ C, 3.3Hz at 190◦ C, and 4.7Hz at
200◦ C for a 3atm. At temperatures higher than 190◦ C pyrex cells can often brown
because the alkali metal reacts with the glass in the cell wall. We can eliminate this
effect by employing aluminosilicate glass instead.
We have also investigated the influence of imperfections in the gyroscope for
e , and C n from
a Rb-Ne co-magnetometer. The spin exchange correction factors Cse
se
eq. (3.4), and eq. (3.5) can be estimated. For operation with the pancake cell
e ≈ 8 × 10−5 , and C n ≈ 3 × 10−4 . We predict a false signal due to misalignment
Cse
se
of the pump-probe orthogonality by 1µrad of ≈ 9 × 10−7 rad/s. The rotational
√
sensitivity of the gyroscope is 1 × 10−9 rad/s/ Hz.
124
Chapter 7
Conclusions and future work
We have been able to measure many of the parameters necessary to construct an
optimized enriched neon co-magnetometer. The neon-neon quadrupolar relaxation rate was found to be 214 ± 10min×amagat. The neon-potassium spin exchange coefficient has measured and is listed tables 7.1. The neon-rubidium spin
exchange rate coefficient has been measured and is listed in Table 7.2. The Fermi
contact interaction κ0 for neon with both species is listed in table 7.3. The spin
destruction cross-sections for each alkali with neon is listed in Table 7.4.
We have demonstrated operation of an co-magnetometer gyroscope with mea√
sured sensitivity of ∆Ω ≈ 5.0 × 10−7 rad/s/ Hz. The fundamental sensitivity of
√
such a gyroscope using the current K-He mixture is ∆Ω ≈ 1.0 × 10−8 rad/s/ Hz
and is limited by angular noise of the probe beam. We have shown that switching
√
to 21 Ne would increase the sensitivity of the detector to ∆Ω ≈ 1.0 × 10−9 rad/s/ Hz.
Spin Exchange Rate Method
Repolarization
Theoretical Prediction
Rate of Rise
cm3 /s
3.36 ± 0.42 × 10−20
7.5 × 10−20
2.34 ± 0.29 × 10−20 cm2
Table 7.1: Spin exchange parameter of a potassium-neon system
125
Spin Exchange Rate Method
Repolarization
Theoretical Prediction
Rate of Rise
cm3 /s
0.80 ± 0.12 × 10−19
1.66 × 10−19
0.82 ± 0.18 × 10−19
Table 7.2: Spin exchange parameter of a rubidium-neon system
Species
K-Ne
K-Ne (prediction)
Rb-Ne
Rb-Ne (Walsworth)
Rb-Ne (prediction)
Fermi Contact Interaction
30.8 ± 2.7
34
35.7 ± 3.7
32.0 ± 2.7
38
Table 7.3: Fermi contact interaction measurements for the K-Ne, and Rb-Ne systems in this work, and compared to both predictions ans measurements by other
groups.
Species
K-Ne
K-Ne
K-Ne
Rb-Ne
Rb-Ne
Rb-Ne
Rb-Ne
Group
This work
Franz
Walker (prediction)
This work
Franz and Volk
Franzen
Walker (prediction)
10−23 cm2
1.1 ± 0.1
1.41 ± 0.14
1.6
1.9 ± 0.2
1.9
5.2
1.8
Table 7.4: Spin destruction cross sections for K-Ne, and Rb-Ne as compared to both
theory and other measurements
126
Switching to 21 Ne will also enable stricter limits on tests of Lorentz and CPT violation.
In future when using
21 Ne
to construct an improved co-magnetometer one
must further study the effects of quadrupole interactions and cell geometry which
differ from those of 3 He. Since the dominant source of relaxation in a neon cell is
due to quadrupolar relaxation, and not due to long range dipolar fields one could
use a cubic geometry for the measurement cell. This will reduce effects of cell birefringence and beam distortion of the lasers on the magnetometer noise. However
in this case the Bloch equations which describe the magnetometer interaction must
be modified to include a neon self interacting term due to the non-uniform magnetic field it would experience. Also the effect of quadrupolar splitting shifts the
Zeeman levels for the neon gas. This splitting in not uniform for each of the Zeeman levels. To counteract this one needs to with operate with a cell which has zero
quadrupole moment. We have found that in order to generate high neon polarization one requires a uniform alkali polarization. One could also imagine creating a
thin pancake shaped cell so that the cell will be uniformly polarized.
The addition of a quadrupole moment allows one to test more parameters of
the standard model than helium. One could test Lorentz invariance by searching for a quadrupole splitting in the Zeeman levels of the
21 Ne
which varies at a
frequency of twice a sidereal day. One could imagine constructing a vapour cell
with both
21 Ne
and 3 He. The
21 Ne
could be used as a magnetometer and used
to correlate any systematic effects experienced by the 3 He . This would give a
test for the local Lorentz invariance by measuring the mass anisotropy of
21 Ne
(Chupp et al. (1989)). One could also measure the coherence relaxation between
the Zeeman levels of 21 Ne. This has implications, and gives a test of the linearity
of quantum mechanics (Chupp & Hoare (1991)). The quadrupole moment of 21 Ne
interacts with electric field gradients. Thus it could have many potential applica-
127
tions in pulmonary imaging. When the 21 Ne interacts with pulmonary cell walls it
could be used as a probe of the surface dynamics. In fact another Noble gas with
a quadrupole moment
83 Kr
has been shown to successfully detect the difference
between hydrophilic and hydrophobic surfaces (Raftery (2006)). Since many lung
disease are dependent on the surface to volume ratio of lung tissue this could be
useful in diagnosis of pulmonary disease. As one can see the future of 21 Ne optical
pumping is bright.
128
Appendix A
Properties of Ne21
129
Property
Natural Abundance
Gyromagnetic ratio
Quadrupole moment
Spin
Nuclear Magnetic Moment
Diffusion Constant coefficient K in Ne
Diffusion Constant coefficient Rb in Ne
Self Diffusion Constant coefficient Ne
Pressure broadening
0.19
Magnitude
0.27
−336.10
0.10155
3/2
−
√0.661796
1+ T/(273.15K )
pn /(1Amg)
√
1+ T/(273.15K )
pn /(1Amg)
0.235
√
1+ T/(273.15K )
0.79
pn /(1Amg)
9.2
Units
%
Hz/G
10−24 cm2
–
µN
cm2 /s
cm2 /s
cm2 /s
GHz/Amg
Table A.1: Properties of Ne21 relevant for optical pumping. The diffusion constant
coefficients are taken from Franz & Volk (1982)for K in Ne, Franz & Volk (1976) for
Rb in Ne, and Weissman (1973) for Ne in Ne
130
Bibliography
Abragam, A. (1961). Princeiples of Nuclear Magnetism. Oxford University Press.
Adrian, F. J. (1965). Quadrupolar relaxation of xe131 in xenon gas. Phys. Rev.,
138(2A), A403–A409.
Alcock, C. B., Itkin, V. P., & Horrigan, M. K. (1984). Vapor-pressure equations for
the metallic elements - 298-2500 k. Canadian Metallurgical Quarterly, 23(3), 309–
313.
Allred, J. C., Lyman, R. N., Kornack, T. W., & Romalis, M. V. (2002).
High-
sensitivity atomic magnetometer unaffected by spin-exchange relaxation. Physical Review Letters, 89(13), 130801.
Anderson, L. W., Pipkin, F. M., & Baird, J. C. (1959). N14 -n15 hyperfine anomaly.
Physical Review, 116(1), 87–98.
Anderson, L. W., Pipkin, F. M., & Baird, J. C. (1960). Hyperfine structure of hydrogen, deuterium, and tritium. Physical Review, 120(4), 1279–1289.
Andronova, I. & Malykin, G. Phys. Usp, 45, 793.
Appelt, S., Baranga, A. B.-A., Erickson, C., Romalis, M., Young, A., & Happer, W.
(1998). Phys.Rev.A., 58, 1412.
131
Appelt, S., Ben-Amar Baranga, A., Erickson, C. J., Romalis, M. V., Young, A. R., &
Happer, W. (1998). Theory of spin-exchange optical pumping of 3 he and 129 xe.
Physical Review A, 58(2), 1412–1439.
Avenel, O., Mukharsky, Y., & Varoquaux, E. (June 2004). Superfluid gyrometers.
Journal of Low Temperature Physics, 135, 745–772(28).
Baracchino, L., Basso, G., Ciofi, C., & Neri, B. (1997).
Ultra-low noise pro-
grammable voltage source. IEEE trans. Instrum. Meas., 46, 1256.
Barton, A. S., Newbury, N. R., Cates, G. D., Driehuys, B., Middleton, H., & Saam, B.
(1994). Self-calibrating measurement of polarization-dependent frequency shifts
from rb−3he collisions. Phys. Rev. A, 49(4), 2766–2770.
Bell, W. E. & Bloom, A. L. (1957). Optical detection of magnetic resonance in alkali
metal vapor. Physical Review, 107(6), 1559–1565.
Bhaskar, N. D., Pietras, J., Camparo, J., Happer, W., & Liran, J. (1980). Spin destruction in collisions between cesium atoms. Phys. Rev. Lett., 44(14), 930–933.
Buchman, S., Everitt, C. W. F., Parkinson, B., Turneaure, J. P., & Keiser, G. M. (2000).
Cryogenic gyroscopes for the relativity mission. Physica B: Condensed Matter,
280(1-4), 497 – 498.
Budker, D., Kimball, D. F., & DeMille, D. P. (2004). Atomic Physics an exploration
through problems and solutions. Oxford University Press.
Canuel, B., Leduc, F., Holleville, D., Gauguet, A., Fils, J., Virdis, A., Clairon, A.,
Dimarcq, N., Bordé, C. J., Landragin, A., & Bouyer, P. (2006). Six-axis inertial
sensor using cold-atom interferometry. Physical Review Letters, 97(1), 010402.
Cates, G. D., Schaefer, S. R., & Happer, W. (1988). Relaxation of spins due to field
132
inhomogeneities in gaseous samples at low magnetic fields and low pressures.
Physical Review A, 37(8), 2877–2885.
Chann, B.and Babcock, E. A. L. & Walker, T. (2002). Measurements of he-3 spinexchange rates. Phys. Rev.A., 66(9), 032703.
Chupp, T. E. & Coulter, K. P. (1985). Polarization of ne21 by spin exchange with
optically pumped rb vapor. Phys. Rev. Lett., 55(10), 1074–1077.
Chupp, T. E. & Hoare, R. J. (1991). Erratum: ‘‘coherence in freely precessing ne21
and a test of linearity of quantum mechanics’’ [phys. rev. lett. 64, 2261 (1990)].
Phys. Rev. Lett., 66(1), 120.
Chupp, T. E., Hoare, R. J., Loveman, R. A., Oteiza, E. R., Richardson, J. M.,
Wagshul, M. E., & Thompson, A. K. (1989). Results of a new test of local lorentz
invariance: A search for mass anisotropy in ne21. Phys. Rev. Lett., 63(15), 1541–
1545.
Clarke, J. & Braginski, A. I. (2004). The SQUID Handbook. Wiley-VCH.
Corney, A. (1977). Atomic and Laser Spectroscopy. Clarendon Press.
Dehmelt, H. G. (1957). Modulation of a light beam by precessing absorbing atoms.
Physical Review, 105(6), 1924–1925.
Erickson, C. J. (2000). Measurements of the magnetic field dependence of the spin relaxation rate in alkali metal vapors. PhD thesis, Princeton University.
Franz, F. A. & Volk, C. (1976). Spin relaxation of rubidium atoms in sudden and
quasimolecular collisions with light-noble-gas atoms. Physical Review A, 14(5),
1711–1728.
Franz, F. A. & Volk, C. (1982). Electronic spin relaxation of the 42 s1/2 state of k
induced by k-he and k-ne collisions. Physical Review A, 26(1), 85–92.
133
Franzen, W. (1959). Spin relaxation of optically aligned rubidium vapor. Physical
Review, 115(4), 850–856.
Franzen, W. & Emslie, A. G. (1957). Atomic orientation by optical pumping. Phys.
Rev., 108(6), 1453–1458.
Fukushima, E. & Roeder, S. B. W. (1981). Experimental NMR: A Nuts and Bolts Approach. Addison-Wesley.
Gardner, W. A. (1990). Introduction to Random Processes. McGraw-Hill.
Gentile, T. R. & McKeown, R. D. (Jan 1993). Spin-polarizing he-3 nuclei with an arclamp-pumped neodymium-doped lanthanum magnesium hexaluminate laser.
Phys.Rev.A, 47(1).
Grover, B. (1983).
Production of a nuclear-spin-polarized neon-21 sample.
Phys.Rev.A., 28(4).
Gustavson, T. L., Bouyer, P., & Kasevich, M. A. (1997). Precision rotation measurements with an atom interferometer gyroscope. Phys. Rev. Lett., 78(11), 2046–2049.
Gustavson, T. L., Landragin, A., & Kasevich, M. A. (2000). Rotation sensing with a
dual atom-interferometer sagnac gyroscope. Classical Quantum Gravity, 17, 2385.
Happer, W. (1972). Optical pumping. Reviews of Modern Physics, 44(2), 169–249.
Happer, W. & Mathur, B. S. (1967). Effective operator formalism in optical pumping. Physical Review, 163(1), 12–25.
Happer, W. & Tam, A. C. (1977). Effect of rapid spin exchange on the magneticresonance spectrum of alkali vapors. Physical Review A, 16(5), 1877–1891.
Herman, R. M. (1965). Theory of spin exchange between optically pumped rubidium and foreign gas nuclei. Phys. Rev., 137(4A), A1062–A1065.
134
Jackson, J. D. (1999). Classical Electrodynamics. John Wiley & Sons.
Jentsch, C., Muller, T., Rasel, E. M., & Ertmer, W. (2004). Relativ.Gravit., 36, 2197.
Kadlecek, S., Anderson, L. W., & Walker, T. G. (1998). Field dependence of spin
relaxation in a dense rb vapor. Phys. Rev. Lett., 80(25), 5512–5515.
Kastler, A. (1957). Optical methods of atomic orientation and of magnetic resonance. Journ.Opt.Soc.Am, 47, 460–465.
Khriplovich, I. B. & Lamoreaux, S. K. (1997). CP Violation Without Strangeness.
Springer.
Knappe, S., Schwindt, P. D. D., Gerginov, V., Shah, V., Liew, L., Moreland, J., Robinson, H. G., Hollberg, L., & Kitching, J. (2006). Microfabricated atomic clocks and
magnetometers. Journal of Optics A, 8(7), S318–S322.
Knappe, S. e. a. (2004). A microfabricated atomic clock. Applied Physics Letters, 85,
1460.
Kornack, T., Ghosh, R., & Romalis, M. (2005a). Nuclear spin gyroscope based on
an atomic co-magnetometer. Phys.Rev.Lett., 95(21).
Kornack, T. W., Ghosh, R. K., & Romalis, M. V. (2005b). Nuclear spin gyroscope
based on an atomic comagnetometer. Physical Review Letters, 95(23), 230801.
Kornack, T. W. & Romalis, M. V. (2002). Dynamics of two overlapping spin ensembles interacting by spin exchange. Physical Review Letters, 89(25), 253002.
Kornack, T. W., Smullin, S. J., Lee, S.-K., & Romalis, M. V. (2007). A low-noise
ferrite magnetic shield. Applied Physics Letters, 90(22), 223501.
135
Kornack, T. W., Vasilakis, G., & Romalis, M. V. (2008). Preliminary results from a
test of cpt and lorentz symmetry using a k-3 he co-magnetometer. In Kostelecký,
V. A. (Ed.), CPT and Lorentz Symmetry IV, (pp. 206–213).
Lwin, N. & McCartan, D. G. (1978). Collision broadening of the potassium resonance lines by noble gas. J.Phys. B:Atom.Molec.Phys., 11(22).
Migdalek, J. & Kim, Y.-K. (1998). Core polarization and oscillator strength ratio
anomaly in potassium, rubidium and caesium. Journal of Physics B, 31(9), 1947–
1960.
Mort, J., Lüty, F., & Brown, F. C. (1965). Faraday rotation and spin-orbit splitting of
the f center in alkali halides. Phys. Rev., 137(2A), A566–A573.
Müller, T., Wendrich, T., Gilowski, M., Jentsch, C., Rasel, E. M., & Ertmer, W. (2007).
Versatile compact atomic source for high-resolution dual atom interferometry.
Physical Review A (Atomic, Molecular, and Optical Physics), 76(6), 063611.
Newbury, N. R., Barton, A. S., Bogorad, P., Cates, G. D., Gatzke, M., Mabuchi, H., &
Saam, B. (1993). Polarization-dependent frequency shifts from rb−3he collisions.
Phys. Rev. A, 48(1), 558–568.
Oros, A. & Shah, N. (2004). Hyperpolarized xenon in nmr and mri. Phys.Med.Biol,
49, R105–R153.
Ottinger, C., Scheps, R., York, G. W., & Gallagher, A. (1975). Broadening of the rb
resonance lines by the noble gases. Phys. Rev. A, 11(6), 1815–1828.
Powles, J. (1958). The adiabatic fast passage experiment in magnetic resonance.
Proc.Phys.Soc, 71, 497–500.
Purcell, E. M. & Field, G. B. (1956). Influence of collisions upon population of
hyperfine states in hydrogen. Astrophysical Journal, 124(3), 542–549.
136
Pustelny, S., Jackson Kimball, D. F., Rochester, S. M., Yashchuk, V. V., & Budker, D.
(2006). Influence of magnetic-field inhomogeneity on nonlinear magneto-optical
resonances. Physical Review A, 74(6), 063406.
Raftery, D. (2006). Magnetic resonance imaging: a cryptic contrast agent. Nature
Physics, 2, 77–78.
Romalis, M. V. & Cates, G. D. (1998). Accurate 3he polarimetry using the rb zeeman
frequency shift due to the rb − 3he spin-exchange collisions. Phys. Rev. A, 58(4),
3004–3011.
Romalis, M. V., Griffith, W. C., Jacobs, J. P., & Fortson, E. N. (2001). New limit on the
permanent electric dipole moment of 199 hg. Phys. Rev. Lett., 86(12), 2505–2508.
Sanders, S. J., Strandjord, L. K., & Mead, D. (2000). Optical Fiber Sensors Conference
Tech. Digest, 5.
Savukov, I. M. & Romalis, M. V. (2005). Effects of spin-exchange collisions in a
high-density alkali-metal vapor in low magnetic fields. Physical Review A, 71(2),
023405.
Savukov, I. M., Seltzer, S. J., Romalis, M. V., & Sauer, K. L. (2005). Tunable atomic
magnetometer for detection of radio-frequency magnetic fields. Physical Review
Letters, 95(6), 063004.
Schaefer, S. R., Cates, G. D., Chien, T.-R., Gonatas, D., Happer, W., & Walker, T. G.
(1989a). Frequency shifts of the magnetic-resonance spectrum of mixtures of
nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal
atoms. Phys. Rev. A, 39(11), 5613–5623.
Schaefer, S. R., Cates, G. D., Chien, T.-R., Gonatas, D., Happer, W., & Walker, T. G.
(1989b). Frequency shifts of the magnetic-resonance spectrum of mixtures of
137
nuclear spin-polarized noble gases and vapors of spin-polarized alkali-metal
atoms. Phys. Rev. A, 39(11), 5613–5623.
Schearer, L. D. (1968). Optical pumping of neon metastable (p23) atoms. Phys. Rev.
Lett., 21(10), 660–661.
Stedman, G. (1997). Ring-laser tests of fundamental physics and geophysics. Rep.
Prog. Phys., 60, 615–688.
Stoner, R. E. & Walsworth, R. L. (2002). Measurement of the 21ne zeeman frequency
shift due to rb − 21ne collisions. Phys. Rev. A, 66(3), 032704.
Sumner, T., Pendlebury, J., & Smith, K. (1987). Conventional magnetic shielding.
J.Phys.D:Appl.Phys, 20, 1095–1101.
Trusov, A. A., Schofield, A. R., & Shkel, A. M. (2008). Performance characterization of a new temperature-robust gain-bandwidth improved mems gyroscope
operated in air. Sensors and Actuators A: Physical, In Press, Corrected Proof, –.
Walker, T. (1989a). Estimates of spin-exchange parameters for alkali-metal-noblegas pairs. Phys.Rev.A., 40(9).
Walker, T. & Happer, W. (1997). Spin-exchange optical pumping of noble-gas nuclei. Rev.Mod.Phys., 69, 629–642.
Walker, T. G. (1989b). Estimates of spin-exchange parameters for alkali-metal—
noble-gas pairs. Physical Review A, 40(9), 4959–4964.
Weissman, S. (1973). Self-diffusion coefficient of neon. Physics of Fluids, 16(9), 1425–
1428.
Woodgate, G. (1989). Elementary Atomic Structure. Oxford University Press.
Woodman, K. F., Franks, P. W., & Richards, M. D. Journal of Navigation, 40, 366.
138
Wu, Z., Walker, T. G., & Happer, W. (1985). Spin-rotation interaction of noble-gas
alkali-metal atom pairs. Phys. Rev. Lett., 54(17), 1921–1924.
Xie, H. & Fedder, G. K. (2003). A drie cmos-mems gyroscope. Sensors, 2, 1413–1418.
Yver-Leduc, F. e. a. J. Opt. B.
139